Theore A. Let n (n 4) be arcoplete inial iersed hypersurface in R n+. 4(n ) Then there exists a constant C (n) = (n 2)n S (n) > 0 such that if kak n d

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1 Gap theores for inial subanifolds in R n+ Lei Ni Departent of atheatics Purdue University West Lafayette, IN atheatics Subject Classification 53C2 53C42 Introduction Let be a copact iersed inial subanifold of diension in the unit S n (). It was proved by J. Sions in [S] that if kak 2 < (n ), where A is the second fundaental 2n 2 for, then is totally geodesic. It was also proved by S. S. Chern,. Do Caro and S. Kobayashi using the oving frae in [C-D-K] later. For the inial subanifolds in R n+ it was proved in [Al] that there is a siilar theore for the volue growth. ore precisely, Allard showed that if, a inial subanifold of R n+ has Euclidean volue growth and the density function (x; r) = Vol( Br(x)) V < +ffi, for soe sall positive 0 ()r nuber ffi, then is totally geodesic. On the other hand, it was shown in [An], [F-C] and [Ty] that when is a inial subanifold (of diension )in R n+, the total scalar curvature R kak dv is closed related to the topology and the orse index of. ore recently it was shown in [S-] that if is a stable inial hypersurface with finite total scalar curvature then is totally geodesic. In this short note we will show that there are soe gap theores for the total scalar curvature. These ay be thought as the analogy of the above entioned Sions' theore for the inial subanifolds in R n+. ore precisely we can show the following result: (We should point out that we do not need the stability assuption, which is essential in the above entioned Shen-hu's result in [S-].)

2 Theore A. Let n (n 4) be arcoplete inial iersed hypersurface in R n+. 4(n ) Then there exists a constant C (n) = (n 2)n S (n) > 0 such that if kak n dv n < C (n); ust be totally geodesic. Here A is the second fundaental for of and S(n) is the constant in the L 2 -Sobolev inequality. (For exaple, using the L -Sobolev constant provided in [-S], one can show that S(n) = is the volue of the unit ball in R n.) 2 4 n+ (2n 2)! n =n (n 2) will be big enough, where! n In the proof of Theore A we need first to show the following result about the ends of inial subanifolds, which can be viewed as a gap theore for the nuber of ends. Theore B. Let ( 3) be a coplete inial iersed subanifold of diension in R n+. Then there exists a constant C 2 () = q S ()>0such that if kak dv <C2 () has only one end. Here, as in Theore A, A is the second fundaental for and S() is the constant in the L 2 -Sobolev inequality. The above gap theore further deonstrates the fact that there is a close relation between the topology of the inial subanifold R and the total scalar curvature kak dv. Concerning our second result, there are a few previous results which we should ention. First it was proved in [F-C], for the inial surface in R 3, that the finiteness of the total scalar curvature iplies finiteness of the orse index as well as finite any ends. Later this stateent was generalized to the high diension inial hypersurfaces in [Ty]. ore recently, using the function theory, [C-S-] proved that if is a stable inial hypersurface in R n+ then has only one end. Our Theore B and related results in the next section conclude that has only one end or finite any ends without assuing that is stable or of codiension one. Very recently, [L-W] proved the Liouville property for inial subanifold, whose density function (x; r) satisfies that (x; r)» μ < 2. Naely they showed that there is no bounded haronic functions on such inial subanifolds. Cobining with the observations in the next section of this paper, as a corollary, we know that has at ost one end if its density function satisfies (x; r)» μ < 2. Therefore one can view their result as a gap type theore for the 2

3 nuber of ends. The constant C and C 2 in our theores are far away fro the sharpest because of the dependency on the L 2 -Sobolev constant on inial subanifolds. It would be very nice if one can find the best constant as Sions' theore in [S] and classify all the inial hypersurfaces for which the equality holds as in [C-D-K]. Finally we should ention that the arguent we used can also sharpen a lower bound estiate for the first eigenvalue of a Schrödinger operator of Li-Yau [L-Y2], therefore sharpens the upper bound for the nuber of bound states obtained in [L-Y2] (cf. rearks before Theore 3.2). Acknowledgeent. suggestions. The author would like to thank Professor Peter Li for helpful 2 Preliinaries We first establish soe basic results on the haronic function theory on inial subanifolds of R n+. Using the bounded haronic functions with finite Dirichlet integral we then show a couple of results concerning the finiteness of the nuber of ends for the inial subanifolds in R n+. Let be a inial subanifold of diension 3 in R n+, p 2 be a point on. Let A be the second fundaental for of and kak be the length of the second fundaental for. Let r p (x) be the extrinsic distance function of R n+ with respect to p and ρ p (x) be the intrinsic distance function. We shall also use the following conventions: ~B p (a) : = the ball of radius a centered at p in R n+. D p (a) : = ~ Bp (a) : B p (a) : = the geodesic ball of radius a centered at p in. The following lea will be very convenient to use in the construction of bounded haronic functions on. Lea 2. Let ( 3) be a -diensional inial subanifold in R n+. Then there exist inial positive Green's function G(x; y) on such that li r y(x)! G(x; y) = 0. Proof. Let D p (a) be as the above. The heat-kernel coparison of Cheng-Li-Yau iplies that H D p(a)(x; y; t)» μ Ha (r y (x);t); 3

4 where μ Ha (r y (x);t) is the Dirichlet heat-kernel of ~ B (a)inr (the ball of radius a centered at origin). This iplies that Taking a!wehave that H D p(a)(x; y; t)» (4ßt) n 2 exp( jr y(x)j 2 ): 4t H(x; y; t)» (4ßt) n 2 exp( jr y(x)j 2 ): 4t Integrating along the tie direction, we have that G(x; y)» ( 2)! jr y (x)j 2 : Reark. One can also construct Green's function directly as in [L-T] by copact exhaustion and applying the Sobolev inequality to prove that the liit exists. By using the heat kernel estiate we can get the upper bound of the Green's function as a consequence. As a corollary of the heat kernel estiate we have the following: Corollary 2.2 (ean-value inequality) (See [C-L-Y], [-S]) Let be a inial subanifold in R n+. Suppose f is a nonnegative subharonic function defined on. Then (2.) f(p)» f(x) da (2.2)! f(p)» f(x) dv! a Dp(a) where! is the volue of the unit ball in R. Proof. The first part proof can be found in [C-L-Y] or [-S]. For the second part we only need to apply the co-area forula and note that jrrj». ore precisely f(p)! a =» a 0 a 0 f(p)! s a f(x) jrrj Dp(a) da ds = f(x) dv: f(x) da ds Applying an arguent of Varopoulos (cf. [V]), one can have Sobolev inequality as a corollary of the heat-kernel estiate of Cheng-Li-Yau. 4

5 Corollary 2.3 (Sobolev Inequality) (See also [-S]). Let be a inial subanifold in R n+, then there exists constant S = S() such that (2.3) 2 ffi 2 2 dv» S() for any copact supported sooth function ffi on. jrffij 2 dv Reark. In fact, ichael and Sion proved a stronger version of Sobolev inequality, so-called L -Sobolev inequality, asfollows: (2.4) ffi dv» S jrffij dv: In fact, they showed that S () = 4+ will be enough to have the above inequality.! = Using this fact, we can have a lower bound for S() in the inequality (2.3). In fact 2 4 S() = + (2 2) will be big enough to have inequality (2.3).! = ( 2) Once we have the inial positive Green's function on we can apply the schee of [L-T2] to construct barrier functions at each end of. Lea 2.4 Let be a inial subanifold in R n+, K ρ be a copact subset in, and let E i be the ends with respect to K, then there exist haronic functions g i on E i which satisfy that g i =;li g x! i (x) =0; jrg i j 2 dv < : E i Proof. This result is essentially proved in [L-T2]. For the sake of copleteness we sketch the proof here. Let g j i (x) be haronic function defined on E i D p (r j ) and satisfies g j i = ; g j p(r j ) E i = 0. We have the estiate g j i (x)» CG(p; x), for soe constant C independent of j. By taking j!weget the function g i, which obviously satisfies the first two identities in the conclusion of Lea 2.4. By the fact that g j i (x) iniizes the Dirichlet integral aong all Lipschitz functions with the sae boundary data we know that R jrg j i j 2 dv is a decreasing sequence of j. This establishes the third property of g i. Using the barrier functions on each end we can construct linearly independent bounded haronic functions as in [L-T2]. 5

6 Lea 2.5 Let be a inial subanifold in R n+, K ρ be a copact subset in, E i be the ends with respect to K, then there exist linearly independent haronic functions u i on which satisfy that li u x! i (x) =;li u i x2ei x!x (x) =0; jru i j 2 dv < : μ2e i E i Proof. For the copleteness we sketch the proof here too. Let u j i function on B p (r j ) satisfying that to be the haronic u j p(r j ) E k =0; for k 6= i, u j p(r j ) E i =: Using the barrier functions we construct in Lea 2.4 and taking j! we have the bounded haronic functions. For the proof of finite Dirichlet integral one can apply Lea.4 of [L-T2]. Proposition 2.6 Let be a inial subanifold in R n+. If has Euclidean volue growth, i. e. Vol(D p (r)) li < r! r then has finite any ends. Proof. By Lea 2.5 we know that the nuber of ends is controlled by the diension of bounded haronic functions with finite Dirichlet integral. In the case when has Euclidean volue growth, cobining with the fact that Vol(Dp(r)) (cf. [Ty]) is a increasing r function of r, we know that has the volue doubling property. Since, by Corollary 2.2, the ean value property holds for the subharonic functions, applying a general theore of Peter Li (cf. Theore in [L2], also [C-]) we know that the polynoial growth haronic function space is of finite diension. In particular, the diension of bounded haronic function space is finite. Therefore has only finite any ends. Reark. By a result of. Anderson (cf. [An]), we know that if has finite total scalar curvature R kak n dv then has at ost Euclidean volue growth. Therefore we have the following corollary, which can also be proved using the scaling arguent and the Groov copactness theore (cf. [G-L-P]): Corollary 2.7 Let be inial subanifold in R n+ with finite total scalar curvature. Then has finite any ends. Siilarly, we can have the following results on the finiteness of the nuber of ends. 6

7 Proposition 2.8 Let be inial subanifold of diension in R n+ and let kak 2 (x) be the square of the length of the second fundaental for. If we have kak 2 (x)» k(ρ(x)), where k(t) R is a nonincreasing continuous function such that 0 ρ n k(ρ) dρ <, then has only finite ends. Proof. As in the previous proposition it suffices to prove that the bounded haronic function space is of finite diension. Since in the case is a inial subanifold in R n+ we have Ricci (x) kak 2 (x) k(ρ(x)), a theore of Li-Ta (See [L-T 2]) says that under the assuption of our proposition the bounded haronic function space is of finite diension. Therefore has only finite any ends. At the end we write the following result for the case when isakähler anifold. Proposition 2.9 Let ( 4), akähler anifold of real diension, beacoplete inial subanifold in R n+, and let kak 2 (x) be the square of the length of the second fundaental for. If kak is of square integrable then has only one end. Proof. Just as what we did before we only need to show that there is no nonconstant haronic function u on such that u has finite Dirichlet integral. First since u has finite Dirichlet integral, by Lea 3. of [L] (see also [G]) we know that u is in fact a pluriharonic function. Let v = jruj 2. Once we know u is a pluriharonic function we can sharpen the Bochner forula for v = jruj to get the following inequality: v jrvj2 kak 2 (x)v: v The interested reader can consult Lea 3.2 of [L] for a proof of the above inequality. Since we also know that there exists positive Green's function on by Lea 2., we can apply Li-Yau's theore (cf. Corollary 2.2 of [L-Y]) to conclude v is zero, i.e u is a constant. Reark. Note that the situation here is totally different fro the coplex diension one case. When is a Rieann surface one can find exaples of inial surfaces with finite total curvature and any ends. On the other hand a known result says that if is a coplete inial iersed surfaces in R 3 with finite total curvature and only one ebedded end then is a plane. 3 Gap theores First we can show the following gap theore on the nuber of ends of inial subanifolds in R n+. 7

8 Theore 3. Let ( 3) be a coplete inial iersed subanifold in R n+, and let S() be the Sobolev constant in Corollary 2.3. If then has only one end. kak dv < C2 () = s S proof. We argue by contradiction. By the construction of last section we learn that if has ore than one end then there exists a nontrivial bounded haronic function u(x) on which has finite total energy. Let f(x) = jruj. The Bochner forula fro [S-Y] yields (3.) f f + kak 2 (x) f 2 jrfj2 : Let ' be a cut-off function such that ( if x 2 Bp (r) '(x) = 0 if x 2 n B p (2r), and jr'j» C ;with C =2; r ultiplying ' 2 on both sides of the above inequality (3.) and integrating by parts we can write jrfj 2 ' 2 dv 2 < rf;r' >f'dv+ kak 2 f 2 ' 2 dv jrfj 2 ' 2 dv: Using Schwartz inequality, for any positive nuber I > 0, we have (3.2) kak 2 f 2 ' 2 dv + I On the other hand, the Sobolev inequality yields f 2 jr'j 2 dv ( I) jrfj 2 ' 2 dv jr(f')j 2 dv S (f') 2 2 dv 2 : Siple calculation together with Schwartz inequality yields (3.3) 2 (II +) jrfj 2 ' 2 dv S (f') 2 2 dv ( + II ) f 2 jr'j 2 dv; 8

9 where II is a positive real nuber which will be chosen later. Cobining (3.2) and (3.3) we have kak 2 f 2 ' 2 dv ( I)S II + 2 ψ (f') 2 2 dv I + I II! f 2 jr'j 2 dv: Now applying Hörder inequality to the left hand side of the above inequality we can have kak dv Finally we have ψ I + I II! 2 (f') 2 2 dv f 2 jr'j 2 dv ( I)S II + ψ ( I)S II + ψ I + Choosing I and II sall enough one can ake ψ ( I)S kak dv II + Then we have ψ I + I II Letting r!we will have! f 2 jr'j 2 dv ffl f 2 2 dv» 0; I II!! 2 (f') 2 2 dv f 2 jr'j 2 dv:! kak 2 dv (f') 2 2 dv : ffl>0: (f') 2 2 dv 2 : which iplies that f 0 and therefore u is a constant function. The contradiction here shows that has at ost one end. Rearks. The first reark we want to ake is that the siilar arguent can give an iproveent of a lower bound estiate of Li-Yau (cf. [L-Y2]) on the eigenvalue of the Schrödinger operator. ore precisely we can have the following results. Let q(x) be a positive function defined on D, a doain in R n. We consider the operator q(x) The lower bound estiate for the first eigenvalue of the above operator proved by Li-Yau concludes that n(n 2) μ (Ω n ) n 2 kqk 4e L n ; 2 9

10 where Ω n is the area of the unit (n )-sphere, e is the Euler nuber. The sae arguent as in the proof of Theore 3. gives a slight iproveent on the above estiate. A direct trace of the calculation can show μ n(n 2) (Ω n ) n 2 kqk 4 L n : 2 The second reark is that we do not know whether the assuption >2is necessary or not. There is a theore by Fischer-Colbrie [F-C] saying that if is a inial surface in R 3 with finite total curvature then has finite orse index. By Huber's theore we also know that has finite any ends. In [C-S-], using the Liouville type theore of Schoen-Yau for the stable inial hypersurfaces they showed that if is a stable hypersurface in R n+ then has only one end. Coparing to their result we neither assue that is a stable nor isahypersurface. Now we begin the proof of Theore A. Naely we will show that if the total scalar curvature is saller than a constant C (n) then the inial hypersurface has to be a hyperplane. We believe that the siilar result also holds for inial subanifolds. Theore 3.2 Let n (n 4) be a coplete inial iersed hypersurface in R n+, and let S(n) be the Sobolev constant in Corollary 2.3. If then has to be a hyperplane. kak n dv n < C (n) = vu u t 4(n ) (n 2)n S Before we prove Theore 3.2, we need the following Lea. Lea 3.3 Let n (n 4) be a coplete inial iersed hypersurface in R n+. If the total scalar curvature R kakn dv is finite then R kakn 2 dv is also finite. Proof. First, by Theore 3.3 of [S-] we know that the rescaling sequences i = f r i g converging soothly to a flat open Rieannian anifold in the sense of Cheeger- Groov, to which we can attach one point O such that = [ O is the union of several hyperplanes through the origin O and O is the only singularity of. At the ean tie, by Corollary 2.7 we know that has only finite any ends. Therefore we only need to show that R E kakn 2 dv is finite for each end E. Because of the convergence of the rescaling sequences f i g we know that the end E, with respect to big enough copact subset is a graph over the tangent space at infinity. Now we can use an estiate of Schoen (Proposition 3 of [Sc]) to get the following unifor estiate of kak (x): kak (x)» C r n ; 0

11 for x p (r) and r>>. On the other hand, by Theore 4. of [An], we have Vol(D p (k+)nd p (k))»c 0 (k+) n ; for soe unifor constant C 0. Therefore we have kak n 2 dv = X k=0 (Dp(k+)nDp(k))» C 3 + C 4 ψ X < : k=0 kak n 2 dv! (k +) n k n(n 2) Now we can prove Theore 3.2. Proof of Theore 3.2. We first need the following sharp version of Sion's inequality due to Sion, Schoen and Yau on the length of the second fundaental for. (3.4) kak kak + kak 4 2 n jrkakj2 : One can consult [S-S-Y] for the proof of a ore general forula. Let ' be a cut-off function as in the proof of Theore 3.. ultiplying kak n 4 ' 2 on both sides of the above inequality and integrating by parts we have that (n 3) + jrkakj 2 kak n 4 ' 2 dv 2 kak n ' 2 dv 2 n Applying Schwartz inequality, we can write (n 3) + 2 n II jrkakj 2 kak n 4 ' 2 dv: jrkakj 2 kak n 4 ' 2 dv» II < rkak;r'>kak n 3 'dv kak n 2 jr'j 2 dv + kak n ' 2 dv; (3.5) for any positive nuber II. On the other hand, direct calculation yields the following inequality, after using the Schwartz inequality. jr(kak n 2 2 ')j 2 dv» ( +I)( n 2 ) 2 jrkakj 2 kak n 4 ' 2 dv kak n 2 jr'j 2 dv; I

12 for any positive nuber I. Cobining (3.5) and and above inequality together we have jr(kak n 2 2 ')j 2 dv» n 2 ( + I)( 2 )2 n 3+ 2 II kak n ' 2 dv n ( + I)( n n )2 kak n 2 jr'j n II dv II + ( + I ) kak n 2 jr'j 2 dv: Applying Sobolev inequality one can write S kak n 2 2 n 2 2n 2n n n 2 ' n 2 dv Now let r!. Use Lea 3.3 we have» n 2 ( + I)( 2 )2 n 3+ 2 II kak n ' 2 dv n 0 n ( + I)( S 2 )2! 2 n 3+ ψ 2 II kak n n dv A n ( + I)( n n )2 kak n 2 jr'j n II dv II + ( + I ) kak n 2 jr'j 2 dv: n 2 kak n n dv» 0: Choosing I and II sall enough one can easily find a positive nuber ffl such that S n 2 ( + I)( 2 )2! 2 n 3+ ψ 2 II kak dv n n ffl>0: n Then we can conclude that kak 0. Therefore is a totally geodesic hyperplane. References [Al] W. K. Allard, First variation of a varifold, Annals of ath. 95(972), [An]. Anderson, The copactification of a inial subanifold in Euclidean space by the Guass ap, Preprint(final version in Dept. of ath, California Institute of Technology, Pasadena, CA 925),

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