5.3. Compactness results for Dirichlet spaces Compactness results for manifolds 39 References 39 sec:intro 1. Introduction We mean by a spectr

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1 CONVERGENCE OF SPECTRAL STRUCTURES: A FUNCTIONAL ANALYTIC THEORY AND ITS APPLICATIONS TO SPECTRAL GEOMETRY KAZUHIRO KUWAE AND TAKASHI SHIOYA Abstract. The purpose of this paper is to present a functional analytic framework of some natural topologies on a family of spectral structures on Hilbert spaces and to study the continuity of the spectral structures induced from the Laplacians on Riemannian manifolds and locally nite graphs. We also consider the spectral structures of metric spaces with Dirichlet forms. Contents 1. Introduction 2 2. Topologies on a family of spectral structures Convergence of Hilbert spaces Convergence of bounded operators Convergence of spectral measures on complex Hilbert spaces Convergence of symmetric bilinear forms Convergence of spectral structures Asymptotic behavior of spectra Convergence of manifolds Preinaries for Lipschitz-Riemannian manifolds Compact-Lipschitz convergence of manifolds Shrinking manifolds Blowing-up Riemannian metrics Degenerating Riemannian metrics Convergence of graphs Graph with simplicial metric Simplicial convergence of graphs Almost polarity of the ends of innite graphs Convergence of metric spaces with Dirichlet forms Measured Gromov-Hausdor topology Preinaries for Dirichlet form 32 Date: August 24, Mathematics Subject Classication. Primary 58G25, 53C23; Secondary 31C25, 31C15. Key words and phrases. spectral structure, convergence of Riemannian manifolds, convergence of graphs, the measured Gromov-Hausdor topology, spectrum of Laplacian, Dirichlet form. 1

2 5.3. Compactness results for Dirichlet spaces Compactness results for manifolds 39 References 39 sec:intro 1. Introduction We mean by a spectral structure on a Hilbert space a compatible set = (A; E; E; ft t g t0 ; fr g >0 ) of an innitesimal generator A, a closed symmetric bilinear form E, a spectral measure E, a strongly continuous contraction semigroup ft t g t>0, and a strongly continuous resolvent fr g >0 on the Hilbert space. An important example of spectral structure is one whose generator is the Friedrichs extension of the Laplacian on a (not necessarily complete) Riemannian manifold. Let X be any locally compact Hausdor space and m any Radon measure on it. Denote by L 2 (X; m) the set of L 2 -functions with respect to m with L 2 -inner product (; ) L 2 (X;m). Then, this is a separable real Hilbert space. Consider a family S of triples (X; m; ), where is any spectral structure on L 2 (X; m). In the rst part (x2) sec:top of this paper, we introduce two natural topologies on S, called the strong and compact topologies (Denition 2.11). defn:spectop We actually dene them on a family of spectral structures on general Hilbert spaces, so that our framework is valid on more general settings, such as for spectral structures on L 2 -dierential forms, L 2 -sections on vector bundles, and so on. The convergence of spectral structures with respect to the strong (compact) topology can be rephrased in terms of the symmetric bilinear forms, resolvent families, semigroups, and spectral measures respectively (Theorem 2.2). thm:convspec We prove the continuity (resp. lower semi-continuity) of the spectra of the generators associated with spectral structures with respect to the compact (resp. strong) topology (Proposition 2.3 prop:semicontspec and Theorem 2.3). thm:cptspectra Let us now mention some history. On a xed Hilbert space, there were some classical studies on convergence of spectral structures associated with the perturbation theory of linear operators (cf. [22, 29]) and also more recent studies with the theory of Dirichlet forms [7, 27]. When X moves, there are no natural identication between L 2 (X; m)'s in general. Nevertheless, we have some asymptotic correspondence between them. Using this, Fukaya [11] Fk:laplace rst studied the continuity of k th eigenvalues for each xed k 2 N of the Laplacian on spaces in the measured Gromov-Hausdor closure of the set of closed Riemannian manifolds with a uniform bound of sectional curvature. When X's are spaces in the measured Gromov-Hausdor closure of Riemannian manifolds with a uniform bound of heat kernels, Kasue-Kumura [19, 20] introduced a natural distance function on S, called the spectral distance, for which the uniformity (in some sense) of heat kernels is essential. 2 Kt:perturb, SgKz:intop DM:Gamma, Ms:compmedia KsKm:specconv, KsKm:specconvII

3 On the other hand, our method is much more general and does not need the integral kernels of the semigroups. In the second part (x3) sec:mfd of this paper, we apply the above functional analytic study on spectral structures to some sorts of shrinking, blowing-up, and degenerating sequences of Riemannian manifolds with their Laplacians. We introduce a new topology on the set of Riemannian manifolds, called the compact-lipschitz topology (Denition 3.1), defn:cptliptop for which the sequences of manifolds as above are all convergent. Here, we emphasize that the manifolds which we study in this paper may not be compact, even nor complete, in particular the Laplacians may have continuous spectra, though the earlier related studies are only in the case of discrete spectra, except some very restrictive cases. One of our main theorems (Theorem 3.1) thm:lip states that if a sequence of Riemannian manifolds M i, i = 1; 2; : : :, compact-lipschitz converges to a Riemannian manifold M whose end is almost polar, then the spectral structure associated with the Laplacian of M i strongly converges to that of M. Here, the almost polarity condition is almost equivalent to the negligibility of the boundary in the sense of [13, 14, 15]. As a simple application to this theorem, we have an asymptotic estimate of spectral gaps of any Riemannian manifold (Corollary 3.2). cor:specgap In the third part (x4), sec:graph we study convergence of (locally nite) graphs. Fj:convgraph, Kn:convgraph We see from the discussion in [10, 18] that if a sequence of nite graphs with metrics converges to a nite graph with a metric in a suitable sense, this implies the compact convergence of the spectral structure associated with the metric (Proposition 4.1). prop:fingraph We then extend this for innite graphs (Theorem 4.1). thm:cptgraph The results are analogous to the previous section x3. sec:mfd We also discuss the almost polarity of the ends of innite graphs (x4.3). ssec:almpolargraph In the nal part (x5), sec:dir we consider a family of metric spaces with Dirichlet forms. Under some uniform condition for the metrics and a uniform bound of Poincare constants, we prove the relative compactness of the set of spectral structures associated with spaces in such a family (Theorems 5.1 thm:dircpt and 5.2), thm:dirstrng results which are extensions of those KsKm:specconv, KsKm:specconvII due to Kasue-Kumura [19, 20]. We have an application to the spectral structures of noncompact complete Riemannian manifolds with a uniform lower bound of Ricci curvature. We prove that the relative compactness of the set of such spectral structures with respect to the strong spectral topology. Gf:form, Gf:special, Gf:Hilbert ssec:hilsp sec:top 2. Topologies on a family of spectral structures 2.1. Convergence of Hilbert spaces. Throughout this paper, let A and B be any directed sets. Denote by L a family of pairs (X; m), where X is any locally compact Hausdor space and m any positive Radon measure on X. We take a net f(x ; m )g 2A of elements of L and an element (X; m) 2 L. 3

4 defn:vaguetop Denition 2.1 (Generalized vague topology). We say that f(x ; m )g 2A converges to (X; m) if there exist an m -measurable map f : D(f )! X from a relatively compact m -measurable subset D(f ) X to X for every 2 A such that Z D(f ) u f dm = Z X u dm for any u 2 C 0 (X), where C 0 (X) denotes the set of continuous functions with compact support in X. This convergence is a generalized notion of the vague convergence on the set of Radon measures over a xed space. We thus call it the generalized vague convergence and its induced topology the generalized vague topology. Remark 2.1. The generalized vague topology on L is so weak that the total measure is not necessarily continuous. Also, if (X ; m ) converges to (X; m), then (X ; m ) converges to the subspace (Y; mj Y ) for any open subset Y X with respect to the generalized vague topology. In particular, the generalized vague topology is not even Hausdor. defn:l2top Denition 2.2 (L 2 -topology). A net fu g 2A of functions with u 2 L 2 (X ; m ) is said to (strongly) L 2 -converges to a function u 2 L 2 (X; m) if f(x ; m )g 2A converges to (X; m) with respect to the generalized vague topology and if there exists a net f~u g 2B of functions in C 0 (X) tending to u in L 2 (X; m) such that k ~u? u k L 2 (X ;m ) = 0; where v := vf on D(f ) and v := 0 on X nd(f ) for v 2 C 0 (X). This convergence denes a topology, say the (strong) L 2 -topology, on the disjoint union L 2 (L) := G (X;m)2L L 2 (X; m): We can also extend this denition to the space of L 2 -dierential forms on manifolds, also to a space of L 2 -sections on vector bundles, and so on. By this reason, it is most convenient to generalize it to general Hilbert spaces and to forget base spaces. Let K := R or C, and let fh g 2N be a family of separable Hilbert spaces over K. We give a family of linear operators ; : C! H, ; 2 N, where C H are dense linear subspaces. Assume that fh g 2N has a (not necessarily Hausdro) topology such that a net fh g 2A converges to an H if and only if, setting H := H ; H := H ; := ; ; C := C for simplicity, we have k uk H = kuk H for any u 2 C. 4

5 lem:basicprop Note that is asymptotically close to a unitary operator, however it is not necessarily injective for large. Throughout this and later sections, we assume that a net fh g 2A converges to an H. Denition 2.3 (Strong topology on H). We say that a net fu g 2A with u 2 H (strongly) converges to a u 2 H if there exists a net f~u g 2B C tending to u in H such that k ~u? u k H = 0: We call the topology on the disjoint union H := F 2N H induced from this convergence the strong topology on H. Remark 2.2. The space H is a generalization of L 2 (L). In fact, setting fh g 2N := fl 2 (X; m)g (X;m)2L, C := C 0 (X), we obtain u! u in L 2 (L) if and only if u! u in H. Lemma 2.1. Let fu g 2A, fv g 2A be two nets of vectors in H with u ; v 2 H, and let u; v 2 H. Then, we have the following: item:basicprop1 (1) u! 0 2 H in H if and only if ku k H! 0. item:basicprop2 (2) If u! u in H, then ku k H! kuk H. item:basicprop3 (3) If u! u and v! v in H, then au + bv! au + bv in H for any a; b 2 K. item:basicprop4 (4) If u! u and v! v in H, then (u ; v ) H! (u; v) H. (5) If ku? v k H! 0 and u! u in H, then v! u in H. item:basicprop (6) If u! u and v! u in H, then ku? v k H! 0. (7) For any w 2 H there exists a net fw g 2A with w 2 H which converges to w in H. eq:basicprop (2.1) Proof. We prove only ( item:basicprop 6), because the proofs of the others are easy. Assume that u! u and v! u in H. Then, there are ~u ; ~v 2 C both tending to u in H such that It follows that k ~u? u k H = k ~v? v k H = 0: k ~u? ~v k H = k (~u? ~v )k H = k~u? ~v k H = 0; which together with ( eq:basicprop 2.1) completes the proof. Corollary 2.1. The strong topology on H is Hausdor if fh g 2N is Hausdor. Proof. We assume that fh g 2N is Hausdor, so that the it of H is only one. Suppose that u 2 H strongly converges to both u; v 2 H. Then, Lemma 2.1( lem:basicprop 3) item:basicprop3 implies that 0 2 H converges to u?v. By Lemma lem:basicprop 2.1( 1), item:basicprop1 we have u? v = 0. This completes the proof. 5

6 defn:weaktop eq:weaktop (2.2) Denition 2.4 (Weak topology on H). We say that a net fu g 2A with u 2 H weakly converges to u 2 H if (u ; v ) H = (u; v) H for any net fv g 2A with v 2 H tending to a v 2 H in H. This convergence induces a topology on H, say the weak topology on H. It is easy to prove that the weak topology on H is Hausdor provided fh g 2N is Hausdor. The weak topology is not stronger than the strong topology on H. Lemma 2.2. Let fu g 2A be a net with u 2 H. If ku k H is uniformly bounded for 2 A, there exists a weakly convergent subnet of fu g 2A. Proof. Let f' k g k2n be a complete orthonormal basis of H. For each k there is a net f ~' k; g 2B C such that ~' k; = ' k in H. Replacing with subnets of A and B if necessarily, we assume that the it (u ; ~' 1; ) H =: a 1 2 R := R [ f?1; 1g exists. Here, it follows from the uniform boundedness of ku k H a 1 6= 1. Repeating this procedure, we may assume that that (u ; ~' k; ) H =: a k 2 R exists for every k 2 N. Let us x a number N 2 N for a while. For any > 0 there is a 2 A such that j ( ~' k; ; ~' l; ) H? kl j < for any k; l = 1; : : : ; N and. Moreover, for any there is an ; 2 A such that j ( ~' k; ; ~' l; ) H? kl j < for any k; l = 1; : : : ; N and ;. Therefore, setting L ; := h ~' k; i k=1;:::;n (the linear span), we have NX (u ; ~' k; ) 2 H? kp L; u k 2 H < N () k=1 for any ; and, where P L : H! L denotes the projection to a linear subspace L H and N () some function depending only on N such that!0 N () = 0. This implies NX k=1 a 2 k = N X k=1 for any N 2 N, so that ku k 2 H < 1 (u ; ~' k; ) 2 H = kp L; u k 2 u := 1X k=1 a k ' k 2 H: We shall prove that some subnet of fu g 2A weakly converges to u. Take any v 2 H and set b k := (' k ; v). By Lemma 2.1( lem:basicprop 6), item:basicprop it suces to 6

7 show that ( eq:weaktop 2.2) holds for some net fv g 2A. Let ~v N := P N Clearly, ~v N 2 C and N!1 ~v N (u ; ~v N ) H = N X k=1 b k=1 k ~' k;. = v (strongly). We have b k (u ; ~' k; ) H = NX k=1 a k b k ; which tends to (u; v) H as N! 1. Thus, there exists a net fn g 2B of natural numbers (slowly) tending to 1 such that ~v N (strongly) and (u ; ~v N ) H = (u; v) H : = v This completes the proof. Lemma 2.3. Let fu g 2A be a net with u 2 H weakly converging to a vector u 2 H. Then we have sup ku k < 1 and kuk H ku k H : Moreover, u! u strongly if and only if kuk H = ku k H : Proof. Suppose sup ku k = 1. fu k g 1 k=1 of fu g such that ku k k Hk Then, there is a countable subnet k. Setting v k := 1 k u k ku k k ; one has kv k k Hk (u k ; v k ) Hk! (u; 0) H = 0. On the other hand, = 1=k! 0 and hence v k! 0 in H, which implies (u k ; v k ) Hk = 1 k ku k k 1 This is a contradiction and thus we obtain sup ku k < 1. Let fv g be a net with v 2 H which strongly converges to u. Then, (u ; v ) H! (u; u) H. Hence, 0 = = This completes the proof. ku? v k 2 H (ku k 2 H? 2(u ; v ) H + kv k 2 H ) ku k 2 H? kuk 2 H: lem:convvec Lemma 2.4. Let u 2 H and let fu g 2A be a net of vectors in H. Then, u! u strongly if and only if (u ; v ) H! (u; v) H for any net fv g 2A of vectors v 2 H weakly tending to a vector v 2 H. Proof. The `only if' part is trivial. We prove the `if' part. The assumption implies that u! u weakly. Setting v := u and u := v in the assumption, we have ku k H! kuk H. This completes the proof. 7

8 2.2. Convergence of bounded operators. Denote by L(H) the set of bounded linear operators from H to H, and by kk L(H) the operator norm. Let B 2 L(H) and B 2 L(H ) for all 2 A. Denition 2.5 (Topologies of operators). We say that fb g 2A strongly (resp. weakly) converges to B if B u! Bu strongly (resp. weakly) for any net fu g 2A with u 2 H strongly (resp. weakly) tending to a u 2 H. We say that fb g 2A compactly converges to B if B u! Bu weakly for any net fu g 2A with u 2 H strongly tending to a u 2 H. The topology induced from the strong (resp. weak) convergence is called the strong (resp. weak) topology on L(H) := F 2N L(H ). The strong and weak topologies are both Hausdor if fh g 2N is Hausdor. It is clear that if B! B compactly, then B! B strongly and weakly. The compact convergence does not induce a topology on L(H) as is seen in Remark rem:cpttop 2.3 below. lem:cmptnorm Lemma 2.5. Assume that H = H (i.e., = ) for all 2 H and that fkb k L(H) g is uniformly bounded. Then, we obtain B! B strongly in L(H) if and only if B! B strongly in L(H). Proof. The lemma follows from kb u? B uk kb k L(H) ku? uk H. lem:convop eq:convop (2.3) Lemma 2.6. Let u; v 2 H be any vectors and fu g 2A ; fv g 2A any nets of vectors with u ; v 2 H. Then we have the following: (1) B! B strongly if and only if (B u ; v ) H = (Bu; v) H for any fu g; fv g; u; v such that u! u strongly and v! v weakly. (2) B! B weakly if and only if ( 2.3) eq:convop holds for any fu g; fv g; u; v such that u! u weakly and v! v strongly. (3) B! B compactly if and only if ( 2.3) eq:convop holds for any fu g; fv g; u; v such that u! u weakly and v! v weakly. Proof. The lemma follows from the denitions of convergences and Lemma lem:convvec 2.4. lem:symweakstrong Lemma 2.7. Assume that B and B are all symmetric. Then, B! B strongly if and only if B! B weakly. Proof. Assume that B! B strongly. Let u; v 2 H and let fu g 2A and fv g 2A be any two nets of vectors in H with u ; v 2 H such that u! u strongly and v! v weakly. Then, since B and B are symmetric and since B u! Bu strongly, we have (u ; B v ) H = (B u ; v ) H! (Bu; v) H = (u; Bv) H : This means that B v! Bv weakly and thus B! B weakly. 8

9 lem:cptop Conversely, we assume that B! B weakly. Let fu g 2A be any net of vectors u 2 H which strongly converges to a vector u 2 H. Then, B u! Bu weakly and so B 2 u! B 2 u weakly. Hence we have kb u k 2 H = (B 2 u ; u ) H! (B 2 u; u) H = kbuk 2 H; which shows that B u! Bu strongly. This completes the proof. Lemma 2.8. Assume that B and B are all symmetric. If B! B compactly, then B is a compact operator. Proof. Let fu g 2B be a net of vectors in H weakly converging to a vector u 2 H. Then, by Lemma 2.7, lem:symweakstrong Bu! Bu H-weakly. It suces to prove that Bu! Bu H-strongly. For each 2 B, there is a net fu ; g 2A with u ; 2 H such that u ; = u H-strongly. It follows that B u ; = Bu H-strongly. The diagonal argument yields that there is a subnet f g 2B of A such that u ; = u H-weakly and j kb u ; k H? kbu k H j = 0: This implies that kbu k H! kbuk H and therefore Bu! Bu H- strongly. rem:cpttop Remark 2.3. If B is a symmetric and noncompact operator, then B := B does not compactly converge to B by Lemma 2.8. lem:cptop Therefore, the compact convergence on L(H) does not induce a topology in general. lem:norm Lemma 2.9. (1) If B! B strongly, then (2) If B! B compactly, then kb k L(H) kbk L(H) : kb k L(H) = kbk L(H) : Proof. (1): For any > 0 there is a unit vector u 2 H such that kbuk H > kbk L(H)?. Find a net fu g 2A with u 2 H strongly converging to u. Note that ku k H! 1. Since B! B strongly, we have kb u k H! kbuk H and therefore, kb k L(H) kb u k H ku k H = kbuk H > kbk L(H)? : This completes the proof of (1). (2): Take a net f g 2A of positive numbers with! 0. For each 2 A, there is a unit vector u 2 H such that j kb u k? kb k L(H) j <. Replacing with a subnet of A, we assume that u weakly converges a vector u 2 H with kuk H 1. Since B u! Bu strongly by the assumption, we have kbk L(H) kbuk H kuk H kbuk H = kb u k H = kb k L(H): 9

10 ssec:speccpx This together with (1) completes the proof Convergence of spectral measures on complex Hilbert spaces. Throughout this section, all Hilbert spaces are assumed to be complex, i.e., K = C. Let A and A be selfadjoint operators on H and H respectively. Denote by R and R their resolvents, and by E and E the corresponding spectral measures respectively. We set i := p?1. thm:speccpx Theorem 2.1. The following are all equivalent: item:cpxsemigp (1) e ita! e ita strongly (resp. compactly) for any t 2 R n f0g. item:cpxresolv (2) Ri! R i strongly (resp. compactly) for any 2 R n f0g. item:cpxcptsupp (3) '(A )! '(A) strongly (resp. compactly) for any continuous function ' : R! C with compact support. item:cpxbdd (4) ' (A )! '(A) strongly (resp. compactly) for any net f' : R! C g of bounded continuous functions uniformly converging item:cpxinterv to a bounded continuous function ' : R! C. (5) E (( ; ])! E(( ; ]) strongly (resp. compactly) for any two real numbers < which are not in the point spectrum of A. item:cpxspecmeas (6) (E u ; v ) H! (Eu; v) H vaguely for any nets fu g 2A ; fv g 2A of vectors u ; v 2 H and any u; v 2 H such that u! u strongly and v! v weakly (resp. u! u weakly and v! v weakly). eq:speccpx (2.4) Proof. It is trivial that ( 4) item:cpxbdd implies ( 1), item:cpxsemigp ( 2), item:cpxresolv and ( 3). item:cpxcptsupp Let us prove ( 1) item:cpxsemigp =) ( 3). item:cpxcptsupp The idea of this proof is essentially due to XI x11.4 of [29]. SgKz:intop Consider the set A of continuous functions ' : R! C such that jxj!1 '(x) = 0 and '(A )! '(A) strongly (resp. compactly). Note that for any bounded continuous functions '; : R! C, we have k'(a)? (A)k sup j'(x)? (x)j; x2r which proves that a uniform it of functions in A is also a function in A. Thus, A is a uniformly closed algebra which is closed under complex conjugation. By ( 1), item:cpxsemigp it contains all ' t, where ' t (x) := e itx for t 2 R n f0g. Since f' t g separates the points on R, the Stone- Weierstrass theorem shows that A contains all continuous functions ' : R! C with jxj!1 '(x) = 0, which implies ( 3). item:cpxcptsupp The same proof as above yields ( 2) item:cpxresolv =) ( 3). item:cpxcptsupp Let u; v 2 H and let fu g 2A ; fv g 2A be two nets of vectors u ; v 2 H such that u! u strongly and v! v weakly (resp. u! u weakly and v! v weakly). Letting a := (E u ; v ) H and a := (Eu; v) H, we have Z 1 ('(A)u; v) H = ' da; a(( ; ]) = ((E( ; ])u; v) H ;?1 10

11 ssec:form lem:exi lem:seccount and also the similar formulas for a ; A ; E (( ; ]). Thus, by Lemma lem:convop 2.6 and by a (R)! a(r), we obtain the equivalence between ( 3){ item:cpxcptsupp 6). ( item:cpxspecmeas Remark 2.4. If one (or all) of the compact convergence conditions of Theorem 2.1 thm:speccpx holds, then, by Lemma 2.8, lem:cptop e ita, R i, and E(( ; ]) are all compact operators, and in particular A has only discrete spectrum Convergence of symmetric bilinear forms. Let a net fh g 2A converge to an H and let ff g 2A be a net of functions F : H! R. We here give the notion of the?-it of F, which was originally de- ned by E. De Giorgi (cf. [7]). DM:Gamma Denition 2.6 (?-it). Setting F := 1 on H nh, we extend it to F : H! R and dene, for u 2 H,?- F (u) := sup U2N (u) inf F (v); v2u?- F (u) := sup inf F (v); U2N (u) v2u where N (u) is a neighborhood system at u on H. It is easy to see that?- F (u) and?- F (u) are both lower semi-continuous functions. Note that f u 2 H j?- F (u) < 1 g H. When?- F (u) =?- F (u), this is indicated by?- F (u) and called the?-it of ff g at u. If the?-it F (u) :=?- F (u) exists at every u 2 H, we say that ff g?-converges to F : H! R and that F is the?-it of ff g. Lemma The net ff g 2A has a countable subnet ff i g i2n such that?- i F i (u) exists for any u 2 H. To prove Lemma 2.10 lem:exi we need the following lemma. We take a H i=1 i t H. countable subnet fh i g i2n of fh g 2A and set H 0 := F 1 Lemma The subspace H 0 of H is second countable. Proof. Let fu j g j2n be a dense countable subset of H. For each j there is u ji 2 H i, i 2 N, tending to u j as i! 1. For any i; j 2 N we set O ij := B(u j ; 1=i) [ 1[ k=i B(u jk ; 1=i) H 0 ; where B(u; r) := f v 2 H j ku? vk H < r g is the metric ball for u 2 H, r > 0. It then obvious that O ij is an open neighborhood of u j. We nd a countable basis O i of open subsets of each H i. Let us prove that fo ij g i;j2n [ S i2n O i is a countable basis of open subsets of H 0. Assume that u 2 O i1 j 1 \ O i2 j 2 \ H. It suces to prove that there are i 3 ; j 3 2 N with u 2 O i3 j 3 O i1 j 1 \O i2 j 2. Find a subsequence fu n g of fu j g tending to u. Since n i ku jpi? u ni k Hi = ku jp? uk H for p = 1; 2, there are two large numbers i 3 ; j 3 2 N such that B(u j3 i; 1=i 3 ) B(u jpi; 1=i p ) 11

12 lem:convform for any i i 3 and p = 1; 2. Therefore, u 2 O i3 j 3 O i1 j 1 \ O i2 j 2. This completes the proof. Proof of Lemma lem:exi By Lemma 2.11 lem:seccount and by Theorem 8.5 of [7], DM:Gamma a countable subnet fh i g i2n has a?-convergent subsequence. This completes the proof. In this section, we assume that all bilinear forms are densely dened, nonnegative, and (Hermitian) symmetric. Let fe g 2A be a net of bilinear forms E on H. We sometimes consider a form E as a function u 7! E(u; u) =: E(u) 2 R by setting E(u) := 1 for u =2 D(E). Lemma If fe g 2A?-converges to a function E : H! R, then E is a closed bilinear form. Proof. To see that E is bilinear, it suces to show that for any u; v 2 H and a > 0, E(u) 0; E(0) = 0; E(au) = a 2 E(u); E(u + v) + E(u? v) = 2(E(u) + E(v)): In fact, these are proved by an easy discussion. The closedness of E follows from the lower semi-continuity of E. This completes the proof. Let E be a closed bilinear form on H and denote the domain by D(). The net fe g 2A?-converges to E if and only if the two following (F1) and (F2) hold: (F1) If a net fu g 2A with u 2 D(E ) strongly converges to a u 2 H in H, then E(u) E (u ): (F2) For any u 2 D(E) there exists a net fu g 2A with u 2 D(E ) which strongly converges to u in H and E(u) E(u ): Denote by F(H) the set of closed bilinear forms on H, 2 N. Proposition 2.1. The?-convergence induces a topology on F(H). It is Hausdor if fh g 2N is Hausdor. Proof. It follows from Lemma 2.12, lem:convform (F1), and (F2) that the?-convergence induces a topology. The Hausdor property is implied by the uniqueness of?- and?-. Denition 2.7 (?-topology). We call the topology induced from the?-convergence on F(H) the?-topology. The following denitions are originally due to Mosco [27]. Ms:compmedia Denition 2.8 (Mosco topology). We say that the net fe g 2A Mosco converges to E if both (F2) and the following (F1') hold: 12

13 (F1') If a net fu g 2A with u 2 H weakly converges to a u 2 H, then E(u) E (u ): Note that Mosco convergent implies?-convergent. It follows that the Mosco convergence induces a topology, say the Mosco topology on F(H). It is Hausdor if fh g 2N is Hausdor. Denition 2.9 (Asymptotic compactness). The net fe g 2A is said to be asymptotically compact if for any net fu g 2A such that u 2 H and (E (u ) + ku k 2 H ) < 1, there exists a strongly convergent subnet of fu g 2A. It is easy to see the following: lem:asymcpt Lemma Assume that fe g 2A is asymptotic compact. Then, fe g 2A?-converges to E if and only if fe g 2A Mosco converges to E. In particular, fe g 2A has a Mosco convergent subnet. sec:convspec Denition We say that E! E compactly if E! E with respect to the Mosco topology and if fe g 2A is asymptotically compact Convergence of spectral structures. Throughout this and the next section, let = (A; E; E; ft t g t0 ; fr g >0 ); = (A ; E ; E ; ft t g t0 ; fr g >0 ) be given spectral structures on H and H respectively, where the generators A and A are all assumed to be nonnegative denite. thm:convspec Theorem 2.2. The following are all equivalent: item:form (1) E! E with respect to the Mosco topology (resp. E! E compactly). item:resolv (2) R! R strongly (resp. compactly) for any > 0. item:semigp (3) Tt! T t strongly (resp. compactly) for any t > 0. item:cptsupp (4) '(A )! '(A) strongly (resp. compactly) for any continuous function ' : R! R with compact support. item:bdd (5) ' (A )! '(A) strongly (resp. compactly) for any net f' : R! Rg of bounded continuous functions uniformly converging to a bounded continuous function ' : R! R. item:interv (6) E (( ; ])! E(( ; ]) strongly (resp. compactly) for any two real numbers < which are not in the point spectrum of A. item:specmeas (7) (E u ; v ) H! (Eu; v) H vaguely for any nets fu g 2A ; fv g 2A of vectors u ; v 2 H and any u; v 2 H such that u! u strongly and v! v weakly (resp. u! u weakly and v! v weakly). Proof. The equivalence between ( 2){( item:resolv 7) item:specmeas is obtained in the same way as in the proof of Theorem 2.1. thm:speccpx 13

14 defn:spectop The equivalence between ( 1) item:form and ( 2) item:resolv for the Mosco/strong topology is proved in the same way as in the proof of Theorem of [27]. Ms:compmedia Let us prove ( 6) item:interv =) ( 1) item:form for the compact topologies. It suces to show the asymptotic compactness of fe g. Assume that (E (u ) + ku k 2 H ) M < 1: sup Replacing with a subnet of A, we assume that u! u weakly. Let > 0 be a number which is not in the point spectrum of A. Since Z 1 Z d(e u ; u ) H 1 1 we have ku k 2 H and hence, by ( 6), item:interv ku k 2 H Z 0 Z 0 d(e ()u ; u ) H E (u ) d(e u ; u ) H + M d(eu; u) H + M kuk2 H + M : Since > 0 can be taken to be arbitrarily large, we obtain ku k H kuk H ; M ; which shows that u! u strongly. Thus, fe g is asymptotically compact. Let us prove ( 1) item:form =) ( 3) item:semigp for the compact topologies. Fix a number t > 0, and let fu g be a net of vectors u 2 H weakly converging to a vector u 2 H. We have already proved Tt by Lemma 2.7, lem:symweakstrong T follows that and! T t strongly. Then, t u! T t u weakly. By setting M := sup ku k H, it E(T t u ) = Z 1 e?2t d(e ()u ; u ) H 0 Z 1 1? e?2t d(e ()u ; u ) H 2t 0 = ku k 2 H? kt t u k 2 H 2t M 2t sup ktt u k M: Hence, the asymptotic compactness of fe g shows that Tt u! T t u strongly by replacing with a subnet of A. This completes the proof. Denition If one (or all) of the conditions in Theorem 2.2 thm:convspec holds, we say that f g 2A strongly (resp. compactly) converges to. The strong (resp. compact) convergence induces a topology on the set of all spectral structures (resp. with compact resolvent) on H, 2 N, say the strong (resp. compact) spectral topology. 14

15 prop:realcpx ssec:spectrum prop:semicontspec Remark 2.5. If! compactly, then R, T t, and E(( ; ]) are all compact operators by Lemma 2.8. lem:cptop Therefore, if the resolvent of is not a compact operator, := cannot compactly converge to. Thus, the compact convergence does not induces a topology on the set of all spectral structure on H, 2 N, in general. For a real Hilbert space H we have the complex Hilbert space ^H := H C. Conversely, for a given complex Hilbert space ^H we have the real Hilbert space H with ^H = H C. There is a 1-1 correspondence between selfadjoint operators A on H and selfadjoint operators ^A on ^H such that ^A(u + iv) = Au + iav for u; v 2 H, or ^Aj H = A. Ofcourse, A is nonnegative if and only if so is ^A. Assume that they are nonnegative. Denote by `^' the complexed object associated with ^A. For example, it follows that for u; v 2 H, ^E(u + iv) = E(u) + E(v), ^Ej H = E, ^E(u + iv) = Eu + iev, and ^Ej H = E. This proves the following: Proposition 2.2. The following are all equivalent: (1)! strongly (resp. compactly). (2) ^! ^ strongly (resp. compactly). (3) One of the conditions in Theorem 2.1 thm:speccpx for ^A; ^A holds Asymptotic behavior of spectra. Denote by () the spectrum of a selfadjoint operator. Proposition 2.3. If! strongly, then (A) (A ); 2A i.e., for any 2 (A) there exist 2 (A ) tending to. Proof. We prove the proposition in the same way as in the proof of Theorem 1.14 in VIII x2 of [22]. Kt:perturb Recalling Proposition 2.2 prop:realcpx and since (A) = ( ^A), we may assume that H and H are all complex Hilbert spaces. Take any 2 (A) and x it. For a number > 0 we set := + i. Then, kr k = 1 inf 2(A) j? j ; and kr k = Hence, by Lemma lem:norm 2.9, inf j? j 2(A ) Since > 0 is arbitrary, this completes the proof. 1 inf 2(A) j? j = 1 : Remark 2.6. The conclusion of Proposition 2.3 prop:semicontspec actually holds even if A and A are not necessarily nonnegative and if one (or all) of the conditions in Theorem 2.1 thm:speccpx holds. 15

16 prop:multi Lemma If two extended real numbers a; b with?1 a < b 1 are both not in the point spectrum of A, we have a E(u) kuk 2 H b for any u 2 E(( a; b ])H n fog. Here we agree that E(( a; b ]) = E(( a; 1)) for b = 1. Proof. Let a < b be not in the point spectrum of A and let u 2 E(( a; b ])H n fog. Then, Z b Z 1 deu = E(( a; b ])u = u = deu a?1 and hence (Eu; u) = 0 on R n ( a; b ). Therefore, if u 2 D(A), Z 1 E(u) = (Au; u) H = d(e()u; u) H =?1 Z b a d(e()u; u) H which is not less than a R b a d(eu; u) H = a kuk 2 H and not greater than b R b a d(eu; u) H = b kuk 2 H. This completes the proof. Dene n(i) := dim E(I)H and n (I) := dim E (I)H for a Borel subset I R. Proposition 2.4. Let a < b be two numbers which are not in the point spectrum of A. If! strongly, we have n (( a; b ]) n(( a; b ]): n(( a;b ]) Proof. Take a complete orthonormal basis f' k gk=1 of E(( a; b ])H. Let n 2 N be any xed number if n(( a; b ]) = 1, and n := n(( a; b ]) if n(( a; b ]) < 1. There are ' k 2 H, k = 1; : : : ; n, with ' k = ' k. Since E (( a; b ])! E(( a; b ]) strongly, we have E (( a; b ])' k = E(( a; b ])' k = ' k and so ke (( a; b ])' kk H = k' k k H = 1: It then follows that inf ku? ' u2e (( a;b ])H kk H = 0; which together with (' k ; '` )H = k` proves that E (( a; b ])H has at least dimension n for all suciently large. This completes the proof. thm:cptspectra Theorem 2.3. Assume that! compactly. Then, for any a; b 2 R n (A) with a < b, we have n (( a; b ]) = n(( a; b ]) for suciently large. In particular, the it set of (A ) coincides with (A). 16

17 Remark 2.7. Under the assumption of this theorem, Lemma lem:cptop 2.8 implies that R, T t, and E(( ; ]) must be all compact operators, so that A has only discrete spectrum and n(( a; b ]) < 1 for any a < b < 1. Proof of Theorem thm:cptspectra 2.3. The spectrum (A) of A is a nite or innite sequence of eigenvalues (0 ) 1 2 n, n 2 f0g [ N [ f1g, with no accumulation, where n = 0 means that it is the empty sequence, and n = 1 means that it is an innite sequence tending to innity. We set L 1 := E ((?1; ])H and L 1 := H for a xed 0 > 0, where = 1 := 1 if n = 0. Let 1 := inff E(u) j kuk H = 1; u 2 L 1 g: Then, since n ((?1; 1? ]) = 0 for any > 0, Proposition prop:multi 2.4 shows 1 1. Therefore, when 1 = 1, we have n = 0 and n (R) = n(r) = 0, which implies the theorem. Suppose 1 < 1. There exist unit vectors ' 1 2 L 1 for all suciently large such that E(' 1 ) = 1. Since E! E compactly and by replacing with a subnet of A, there exists a strong it ' 1 := ' 2 L 1 with E(' 1 ) 1. It follows that k' 1 k H = 1 and 1 = inff E(u) j u 2 L 1 ; kuk H = 1 g E(' 1 ) 1 < 1. Thus, we obtain n 1, 1 = 1 = E(' 1 ), and that ' 1 is a unit eigenvector for 1 of A. We next set L 2 := E ((?1; ])H \ h' 1 i?, L 2 := h' 1 i?, and 2 := inff E(u) j kuk H = 1; u 2 L 2 g: Then, since n ((?1; 2? ]) is equal to zero if 1 = 2, and equal to 1 for any with 0 < < 2? 1, Proposition 2.4 prop:multi shows 2 2. Therefore, when 2 = 1, we have the theorem. Suppose 1 < 1. We take unit vectors ' 2 2 L 2 such that E(' 2 ) = 2. Then, the same discussion as above yields n 2, 2 = 2, and that ' 2 strongly converges to a unit eigenvector ' 2 for 2 of A, by replacing with a subnet of A. Now, let a; b 2 [ 0; 1 ) n (A) be two given numbers with a < b, and K 2 N a number with b < K. We repeat the above discussion up to k = k, k = 1; : : : ; K. It then follows that n (( a; b ]) coincides with the number of k with a < k b, which converges to the number of k with a < k b, namely n(( a; b ]). This completes the proof. Corollary 2.2. Assume that! compactly and that the resolvents R are all compact. Let k (resp. k ) be the kth eigenvalue of A (resp. A ) with multiplicity, and f' dim H kgk=1 an orthonormal basis on H such that ' k is an eigenvector for k of A. When dim H > dim H, we set k := 1 for all k dim H + 1. Then we have k = k for any k. 17

18 Moreover, by replacing with a subnet of A if necessarily, for each xed k 2 N, ' k strongly converges to some eigenvector ' k for k of A such that f' k g 1 k=1 is a complete orthonormal basis on H. Proof. In the proof of Theorem 2.3, thm:cptspectra we can choose ' k as in the statement of the corollary. Thus, the corollary follows from the discussion in the proof of Theorem 2.3. thm:cptspectra ssec:lrmfd sec:mfd 3. Convergence of manifolds 3.1. Preinaries for Lipschitz-Riemannian manifolds. A Lipschitz manifold is dened to be a paracompact topological manifold with atlas f(u ; ' )g 2 whose chart transformations ' '?1 : ' (U \ U )! ' (U \ U ) are bi-lipschitz maps between open subsets of R n. A Riemannian metric on a Lipschitz manifold with atlas f(u ; ' )g 2 is dened to be a family of measurable Riemannian metrics g on ' (U ) R n for all 2 which satises the two following conditions: (1) the compatibility condition under chart transformations (' '?1 ) g = g a:e: (2) for each 2 there exists a constant c 2 ( 0; 1 ) such that c k!k L 2 (R n ) k!k L 2 (g ) c?1 k!k L 2 (R n ) for any C 1 dierential 1-form! on R n with compact support in ' (U ), where kk L 2 (R n ) and kk L 2 (g ) denote the L 2 -norms with respect to the Euclidean metric and g respectively. Note that (2) is required to dene the L 2 -norm of dierential 1-form on the manifold. We also dene a Lipschitz manifold with boundary in an ordinary manner. Let M be a Lipschitz-Riemannian manifold (possibly with boundary), i.e., a Lipschitz manifold equipped with a Riemannian metric. Then, we have the distance function on M induced from the Riemannian metric (see [8]). DP:distLip We dene the (1; 2)-Sobolev space W 1;2 (M) to be the set of L 2 -functions u on M admitting L 2 weak derivatives du. A canonical Dirichlet form E : W 1;2 (M) W 1;2 (M)! R is dened by E(u; v) := Z M hdu; dvi M dvol M ; u; v 2 W 1;2 (M); where h; i M and dvol M respectively denote the inner product and volume measure induced from the Riemannian metric on M. Then, the Hilbert inner product (resp. norm) on W 1;2 (M) is written as (; ) W 1;2 = (; ) L 2 + E(; ) (resp. k k 2 W = k k 2 1;2 L + E()). Denote by W 1;2 2 0 (M) the W 1;2 -closure of the set of W 1;2 -functions with compact support in M. Note that the set of Lipschitz functions with compact support on M is dense in W 1;2 0 (M) and also in C 0 (M) with respect to the uniform norm. The canonical spectral structure (M) on M is dened to be 18

19 ssec:lip prop:lip that induced from the symmetric bilinear form (E; W 1;2 0 (M)), where we mean by (E; F) the symmetric bilinear form E with domain restricted on F W 1;2 (M). Note that (E; W 1;2 0 (M)) is a strongly local regular Dirichlet form in the sense of the abstract Dirichlet form theory (see x5.2). ssec:predir We call its innitesimal generator the Laplacian M on M. Remark that if M is a smooth Riemannian manifold, W 1;2 0 (M) coincides with the W 1;2 -closure of the set C 1 0 (M) of smooth functions with compact support in M and the Laplacian M dened here is the Friedrichs extension of the (ordinary) Laplacian dened on C 1 0 (M). Let M and N be two Lipschitz-Riemannian manifolds. By using the short-time asymptotic formula for heat kernel ([28]), Nr:heat it is easy to prove that if a measurable map f : M! N with f vol N = vol M induces an isomorphism between the spectral structures of M and N, i.e., D( M ) = f u f j u 2 D( N ) g and M (u f) = ( N u) f for any u 2 D( N ), then M and N are isometric. In other words, each isomorphism class of triples (X; m; ) for (X; m) 2 L and a spectral structure on L 2 (X; m) contains at most one triple (M; vol M ; (M)) induced from a Lipschitz-Riemannian manifold M. An -almost isometry from M to N is dened to be a bi-lipschitz map f : M! N with j ln dil(f)j + j ln dil(f?1 )j, where dil(f) denotes the dilatation of f, i.e., the smallest Lipschitz constant of f. The Lipschitz distance d L (M; N) between M and N is dened to be the inmum of > 0 such that an -almost isometry from M to N exists. The topology on the set of isometry classes of Lipschitz-Riemannian manifolds induced from the Lipschitz distance d L is called the Lipschitz topology Compact-Lipschitz convergence of manifolds. Let us rst present a trivial proposition. Proposition 3.1. Let fm g 2A be a net of Lipschitz-Riemannian manifolds and M a Lipschitz-Riemannian manifold which is covered by a nitely many local charts. If fm g 2A converges to M with respect to the Lipschitz topology, then (M ; vol M ) converges to (M; vol M ) with respect to the generalized vague topology, and the spectral structure (M ) compactly converges to (M). Consequently, the spectrum ( M ) converges to ( M ). Proof. By the denition of the Lipschitz topology, there is a net of - almost isometries f : M! M. It is clear that (M ; vol M ) converges to (M; vol M ) with respect to the generalized vague topology. In order to prove the strong convergence of spectral structures, it suces to show that the canonical Dirichlet forms E compactly converges to the canonical Dirichlet form E. The pull-back f : W 1;2 (M)! W 1;2 (M ) is a ( )-almost isometry with respect to the W 1;2 -metric, where is some function such that!0 () = 0. Thus, we easily verify (F1) and (F2) in x2.4. ssec:form The asymptotic compactness of fe g follows from the 19

20 compactness of the embedding of W 1;2 (M) into L 2 (M). This completes the proof. The purpose of this section is to extend the above proposition for some sorts of blowing-up, shrinking, and degenerating sequences of manifolds. There, the manifolds are not necessarily complete and so the Laplacians may have continuous spectrum. Since the Lipschitz topology is not suitable for our purpose, we here dene a new notion of convergence of manifolds. defn:cptliptop thm:lip Denition 3.1 (Compact-Lipschitz topology). We say that a net fm g 2A of Lipschitz-Riemannian manifolds compact-lipschitz converges to a Lipschitz-Riemannian manifold M if for any relatively compact open subset O M there exists a net of relatively compact open subsets O M such that O Lipschitz converges to O. This convergence induces a topology on the set of Lipschitz-Riemannian manifolds, say the compact-lipschitz topology. Remark 3.1. Let M be a Lipschitz-Riemannian manifold and fm g 2A a net of Lipschitz-Riemannian manifolds. (1) If M Lipschitz converges to M, then M compact-lipschitz converge to M. However, the converse is not true in general. (2) If M is isometrically embedded onto an open subset of each M, then M compact-lipschitz converges to M. In particular, the compact-lipschitz topology is not Hausdor. (3) If S fm g 2A is an increasing net of open subsets of M with M = M, then M compact-lipschitz converges to M. Denote by X the completetion of a metric space X and call End X := X n X the end of X. For a Lipschitz-Riemannian manifold M, we say that the end End M is almost polar if W 1;2 (M) = W 1;2 0 (M). Theorem 3.1. Assume that a net fm g 2A of Lipschitz-Riemannian manifolds compact-lipschitz converges to a Lipschitz-Riemannian manifold M, and that End M is almost poler. Then, (M ; vol M ) converges to (M; vol M ) with respect to the generalized vague topology, and the spectral structure (M ) on L 2 (M ) strongly converges to (M) on L 2 (M). In particular, we have ( M ) 2A ( M ): Proof. The denition of the compact-lipschitz convergence leads to the existence of -almost isometries f : M D! f(d ) M,! 0, such that D and f(d ) for each are both relatively compact open subsets and ff(d )g is a monotone increasing net covering M. It is easy to verify that (M ; vol M ) converges to (M; vol M ) with respect to the generalized vague topology. We shall show that the canonical Dirichlet form E on L 2 (M ) converges to the canonical Dirichlet form 20

21 E on L 2 (M) with respect to the Mosco topology. To verify (F1') in x2.4, ssec:form we take a net u 2 W 1;2 (M ) with R := ku k W 1;2 (M ) < 1: For a compact subset C M there exists C 2 A such that C f(d ) for all C. Setting u := u f?1 j C 2 W 1;2 (C) for C, we have ku k W 1;2 (C) R: Hence, by replacing with a subnet, for each C, fu g 2A W 1;2 -weakly and L 2 -strongly converges to some u C 2 W 1;2 (C) with ku C k W 1;2 (C) R. We take a monotone increasing sequence of C covering M and use the diagonal argument to obtain a function u 2 W 1;2 (M) such that, (L 2 -strongly) for when replacing with a subnet of A, u = uj C each C, and that E(u) R. This implies (F1'). To verify (F2), let us take any u 2 W 1;2 (M) and x it. Since End M is poler, for any > 0 there is a Lipschitz function ~u in C 0 (M) such that ku? ~u k W 1;2 <. There exists a large such that C := supp ~u f (D ) for all and that the function u 2 W 1;2 (M ) dened by u := ~u f j C on C and by u := 0 outside C satises j E(u )? E(~u )j < and j ku k L 2 (M )? k~u k L 2 (M) j < : Then, u! u in L 2 (L) and E(u )! E(u) as! 0. This completes the proof of the strong convergence of the spectral structures. The rest follows from Proposition 2.3. prop:semicontspec By looking at Theorem 3.1, thm:lip it is important to investigate the criteria of the almost polarity of end. We know the following sucient conditions for it: M is a complete Lipschitz-Riemannian manifold (or End M = ;), which is obtained in a standard way (see [13, 14, 15])). M is a smooth Riemannian manifold and End M is a closed LT:heat, Ms:essential smooth submanifold of M with codimension 2 ([25, 26]). M is an Alexandrov space of curvature bounded below and End M contains no boundary point of M ([24]). KMS:lap M is a Lipschitz-Riemannian manifold and End M has Mincowski codimension > 2, which is obtained by the same way as in [26]. Ms:essential Note that for a C 1 -Riemannian manifold, the almost polarity of the end implies the essential selfadjointness of the Laplacian acting on C 1 0 (M) (see [26]). Ms:essential Remark 3.2. We can easily generalize Proposition 3.1 prop:lip and Theorem thm:lip 3.1 for L 2 -dierential forms instead of L 2 -functions, where the almost polarity of end should be dened for dierential forms. We do not explore it in this paper, because we at present know only few about the almost polarity of end for dierential forms. 21 Gf:form, Gf:special, Gf:Hilbert

22 ssec:shirinking ssec:blowup cor:blowup ex:magnify cor:specgap ssec:degen 3.3. Shrinking manifolds. Let M be a smooth Riemannian manifold whose end End M is almost polar, and N M a closed smooth submanifold of codimension 2. It then follows that W 1;2 0 (M nn) = W 1;2 0 (M) and so (M n N) = (M). Assume that a smooth Riemannian manifold M r, r > 0, contains an open subset U r such that M r nu r is isometric to M n B(N; r). Then, as r! 0, M r compact-lipschitz converges to M n N. Thus, by Theorem 3.1, thm:lip the spectral structure (M r ) strongly converges to (M) as r! Blowing-up Riemannian metrics. Let M be a smooth manifold and S M a closed subset. When a net fg g of Riemannian metrics on M converges to a complete Riemannian metric g on M n S uniformly on compact sets on M n S, we say that g (resp. (M; g )) blows up to g (resp. (M n S; g)). Corollary 3.1. If (M; g ) blows up to (M ns; g), the spectral structure (M; g ) strongly converges to (M n S; g). Proof. It is easy to show that (M; g ) compact-lipschitz converges to (M n S; g). Since M is complete and by Theorem 3.1, thm:lip this completes the proof. Example 3.1. Let M be a smooth Riemannian manifold with metric g, and rm denote the manifold M with metric r 2 g, r > 0. Then, as r! 1, rm blows up to R n, where n := dim M. Therefore, (rm) strongly converges to (R n ) of R n. Corollary 3.2. Let M be a smooth Riemannian manifold. Then we have a a;b!1 b = 1; where ( a; b ) [ 0; 1 ) is any spectral gap, i.e., ( a; b ) \ ( M ) = ;. Proof. As is seen in Example 3.1, ex:magnify the spectral structure (rm) strongly converges to (R n ). Therefore, Theorem 3.1 thm:lip implies that ( rm ) = r?2 ( M ) converges to (R n ) = [ 0; 1 ). This proves the corollary Degenerating Riemannian metrics. Let us next consider degenerating of metrics on a manifold. This is not as easy as shrinking and blowing-up. Let M be a smooth manifold and g a degenerate Riemannian metric on M, i.e., a positive semi-denite smooth (0; 2)- tensor. Then, g induces a pseudo-distance function d g on M and the quotient space of M modulo the equivalence relation d g (; ) = 0 becomes a complete locally compact geodesic space, say ^M g. Here, a geodesic space is a metric space any two points of which can be joined by a length minimizing curve. Denote by g : M! ^M g the projection. The nondegenerate part, say M g, of g in M is an open subset of M and the degenerate part, say S g, of g in M is closed. We equip M g with the 22

23 metric g, so that M g is isometrically embedded into ^M g. With the notation as before, the end End Mg of the completion of M g is identied with the topological g (S g )) of g (S g ) in ^M g. Assume that a net fg g of Riemannian metrics converges to g uniformly on compact sets on M. Then, it is easy to prove that (M; p; g ) converges to ( ^M g ; g (p)) with respect to the pointed Gromov-Hausdor topology (see Denition defn:ghtop 5.1 below for the denition of the pointed Gromov-Hausdor topology). It is also easy to show that (M; g ) compact-lipschitz converges to M g. Let us rst see a typical example. Example 3.2. Let M := R N for an (n? 1)-dimensional closed manifold N, and let fg k g k2n be a sequence of smooth Riemannian metrics on M expressed as dr 2 + f k (r) ds 2 N, where r is the parameter of R, ds 2 N a Riemannian metric on N, and f k : R! R are positive smooth functions. Assume that f k converges uniformly on compact sets to a nonnegative smooth function f : R! R such that f > 0 on R n [ a; b ] and f = 0 on [ a; b ] for some numbers a b. Then, g k converges uniformly on compact sets to a degenerate metric g expressed as dr 2 +f(r) ds 2 N. By Theorem 3.2 thm:degen below, the spectral structure (M; g k ) strongly converges to (M g ), where M g = ((R n [ a; b ]) N; g). For more concrete, we here put f(r) := r 2 and f k (r) := r 2 + 1=k. In this case, ^M g is isometric to the Euclidean cone over the union of two disjoint copies of N (see BGP [2] for the denition of Euclidean cone). It should be remarked that S Mg (or S g ) is not necessarily almost polar as seen in the following: Example 3.3. Let f : R! [ 0; 1 ) be a smooth function and let S := f (x; y) 2 R 2 j 0 y f(x) g. Then, there exists a smooth degenerate Riemannian metric g on R 2 such that Ker g = f v 2 T p R 2 j p 2 S and v is parallel to the y-axisg; where Ker g denotes the set of all v 2 T R 2 with g-norm jvj g = 0. We obtain S = S g, R 2 g = (R 2 n S; g), and S R 2 g Here, the is not almost polar because the one-dimensional Hausdor measure with respect to d g is positive. Let us provide some sort of degenerate metrics with almost polar degenerate set in the following. For 0 m n, let D be the m- dimensional open disk and N a (not necessarily connected) (n? m)- dimensional closed submanifold of a smooth manifold M. Then, the embedded image of N in M has an open neighborhood dieomorphic to the bre (or normal) bundle, say DN, over N with bre D, so that we identify the neighborhood in M with DN. The tangent bundle T (DN) over DN splits as the direct sum of the horizontal sub-bundle H(DN) (which is transversal to the bres) and the vertical sub-bundle V (DN) (which is tangent to the bres). Assume that the structure group G of 23