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1 CADERNOS DE MATEMÁTICA 6, 43 6 May (25) ARTIGO NÚMERO SMA#9 On the essential spectrum of the Laplacian and drifted Laplacian on smooth metric measure spaces Leonardo Silvares Departamento de Matemática e Estatística, CCET, Universidade Federal do Estado do Rio de Janeiro, Av. Pasteur 458, CEP , Rio de Janeiro RJ, Brazil. leo.silvares@uniriotec.br In this paper, we study the L 2 essential spectra of the Laplacian and the drifted Laplacian f on a complete smooth measure metric space (M, g, e f d vol g). Assuming that the Bakry-Émery Ricci curvature tensor satisfies Ric f 2 g and f 2 f in an end E of M, we show that the essential spectrum of the Laplacian is [, ). In such conditions, we proved that the volume E does not grow nor decay exponentially. When Ric f is nonnegative and f has sublinear growth we show that the essential spectrum of the Laplacian is also [, ). May, 25 ICMC-USP. INTRODUCTION Let M be a complete Riemmanian manifold. The Laplace-Beltrami operator = div acting on the space of smooth spaces on M is essentially self-adjoint, thus it may be extended to an unbounded operator, also denoted by, on the Hilbert space L 2 (M). The study and computation of the spectra of the operator on complete non-compact manifolds has been an interest problem in geometric analysis, since these spectra may lead to important geometric and topological information of the manifold. Indeed, in the past two decades, that was a very active area of mathematical research, requiring the use of techniques that belongs to Geometric and Functional Analysis among others. When the manifold has a soul whose exponential map is a diffeomorphism, supposing non-negative sectional curvature and some other additional conditions, Escobar [6] and Escobar-Freire [7] proved that the L 2 spectrum of the Laplacian is [, ). Subsequently, Zhou [6] proved this result with no need to suppose those aditional conditions. Assuming that the manifold has a pole, and supposing the Ricci curvature non-negative, Li [8] proved that the essential spectrum of the Laplacian is [, ). Supposing that the sectional curvature in the radial directions are non-negative rather than the hypothesis of the Ricci curvature, Chen-Lu [4] proved the same result. This result was also proved by Donnelly [5] when the Ricci curvature is non-negative and the volume has Euclidean growth. 43 Sob a supervisão CPq/ICMC

2 44 LEONARDO SILVARES J. Wang [4] removed the need to suppose the existence of a pole, proving that the essential spectrum L p of the Laplacian is [, ), for p [, ), provided the Ricci curvature is bounded below by δ/r 2, where r is the distance from a fixed point on the manifold and δ > depends only on the dimension. Lu and Zhou [9] generalized this result, supposing that lim r Ric =. An important argument in both works was a result proved by Sturm [3] which asserted that the L p essential spectra of the Laplacian are the same, regardless of the value of p [, + ), when the volume has uniform sub-exponential growth and the Ricci curvature is bounded below. By proving a generalization of the Classical Weyl s Criterion, Charalambous and Lu [3] extended the results previously obtained in L 2 to the case where Ric is asymptotically non-negative and lim r r. This generalization of Weyl s Criterion will be used in the results of our paper. Note that most of the above work relates the spectrum of the Laplacian with the Ricci curvature. This is somehow natural, since the Laplacian and the Ricci curvature are related by the Bochner formula 2 u 2 = Hess u 2 + u, u + Ric ( u, u). Indeed, much of the analysis of the Laplacian passes by this formula. There are also many relations between Laplacian and the standard measure of the manifold, that is, the one associated to the volume form. As exemple, we may cite the Green s formulae, the Laplacian as derivative of Dirichlet energy functional, among others. In this paper, we generalized some of these previous results to the context of a smooth metric measure space, that is, a manifold M endowed with a weighted measure of the form e f d vol g, where f is a smooth function, called weight function. The operator f, defined by f u = u f, u, is associated with volume form e f d vol g the same way is associated to d vol g. Moreover, f is a self-adjoint operator on the space L 2 f of square integrable functions on M with respect to the measure e f d vol g. And, since 2 f u 2 = Hess u 2 + u, f u + Ric ( u, u) + Hess f ( u, u), it is natural to define and obtain a correspondent Bochner formula Ric f = Ric + Hess f, 2 f u 2 = Hess u 2 + u, f u + Ric f ( u, u). The operator f is the drifted Laplacian (or drifting Laplacian), and Ric f is the Bakry- Émery Ricci curvature tensor. They are extensions of and Ric, since these ones are Sob a supervisão da CPq/ICMC

3 ON THE ESSENTIAL SPECTRUM OF THE LAPLACIAN AND DRIFTED LAPLACIAN 45 obtained when f is a constant function. The drifted Laplacian is also related with diffusion processes, as studied by Bakry and Émery []. Many results obtained when the Ricci curvature is assumed to be bounded below, such as volume estimates and splitting theorems, have been generalized to the case where the Bakry-Émery Ricci curvature is bounded below. In all these results, some hypotheses are assumed on the growth of weight function f. Interesting generalizations may be found in [5] and []. The Bakry-Émery Ricci curvature plays an important role in the study of the Ricci flow. Gradient Ricci solitons are defined to be complete manifolds (M, g) where Ric f = λg for some λ R, and they are singularities of the Ricci flow. Note that Einstein manifolds are special cases of gradient Ricci solitons, obtained for f c R. Gradient Ricci solitons may be shrinking if λ >, steady if λ =, or expanding if λ < ; and function f is called potential function. The study of such manifolds has been an important motivation to the research on the smooth metric spaces and their related drifted Laplacian and Bakry-Émery Ricci curvature. A remarkable result on shrinking solitons is that the potential function f has quadratic growth. This was proved by Cao and Zhou in [2] and has many consequences on the study of such solitons. In this work, we obtain a similar result in the hypotheses of Theorems.2 and.3 bellow. In the doctoral thesis [2] that led to this work, we studied the L 2 and L 2 f essential spectra of and f respectively, supposing some lower bounds for Ric f and imposing conditions on f. Two of the results we obtained and which were publised in [] are: Theorem.. Let M be a non-compact complete manifold. If f : M R is a smooth f function such that Ric f and lim r r =, the L2 f (M) essential spectrum of f is [, ). (We are denoting by r the distance r(x) = d(p, x) to a fixed point p M.) Theorem.2. Let M be a non-compact complete manifold. If f : M R is a smooth function such that Ric f 2 g and f 2 f, the L 2 (M) essential spectrum of is [, ). If Ric f equation = 2g in Theorem.2, we have a gradient shrinking Ricci soliton, since the R ij + i j f = ρg ij, ρ >, can be normalized to ρ = 2. In this case, Lu - Zhou [9] and Charalambous - Lu [3] proved that the L 2 (M) essential spectrum of is [, ). In an extension to Theorem.2, also present in [2], we proved that its hypotheses need only to be verified in an end of the manifold: Theorem.3. Let M be a non-compact complete manifold. If f : M R is a smooth function such that Ric f 2 g and f 2 f hold on some end of M, the L 2 (M) essential spectrum of is [, + ). Sob a supervisão CPq/ICMC

4 46 LEONARDO SILVARES To prove these results, we needed to obtain some estimates on the f-volume growth. Theses estimates may be of independent interest and are stated in section NOTATION AND BASIC FACTS Let M be a complete noncompact Riemmanian manifold. Given R M and f C (M), we denote denote L p (R) := L p (R, g, d vol g ) and L p f (R) := Lp (R, g, e f d vol g ) and define the norms u L p (R) := u L p (R,g,d vol g) := ( R u p d vol g ) p, ) u L p f (R) := u L p (R,g,e f d vol := g) u (R p e f p d vol g. The spaces L 2 (M) and L 2 f (M) provided of the inner products (u, v) = M uvd vol g and (u, v) f = M uve f d vol g, respectively, are Hilbert spaces. The operators and f defined in may be uniquely extended to (unbounded) self-adjoint operators in L 2 (M) and L 2 f (M), respectively. A point λ C is called a regular point of if ( λ) exists and is a bounded linear operator. If λ C is not a regular point, λ is said to be a spectrum point, and we define the spectrum of, denoted by σ( ), the set of all spectrum points of. We say λ is in the discrete spectrum of if it is an isolated point in σ( ) and it is an eigenvalue of of finite multiplicity. We will denote by σ disc ( ) the discrete spectrum of. The essential spectrum of, denoted by σ ess ( ), is the complementary of σ disc ( ) in σ( ), that is, σ ess ( ) = σ( ) \ σ disc ( ). The definitions of σ( f ), σ disc ( f ) and σ ess ( f ) are analogous. Notice that, since and f are positive and symmetric, their spectra are contained in [, ). Given the manifold M and the function f as above, σ ess ( ) and σ ess ( f ) may be essentially different, as we may see in the following example. Example 2.. Consider the Gaussian soliton ( R n, δ ij, e f(x) d vol g ), where f(x) = x 2 4. For k =,,..., and x i R, we define the Hermite polynomial H k by 2 i H k (x i ) = ( 2) k e x 4 d k dx k i e x 2 i 4. Writing x = (x,..., x n ) R n, and given k,..., k n in {,, 2,...}, we define Sob a supervisão da CPq/ICMC H k,...,k n (x) = H k (x )... H kn (x n ).

5 ON THE ESSENTIAL SPECTRUM OF THE LAPLACIAN AND DRIFTED LAPLACIAN 47 These polynomials are, up to multiplication by a constant, the classical Hermite polynomials of probability. It may be proved that {H k,...,k n } is an orthogonal basis of L 2 (R n, e x2 /4 ). Moreover, a direct calculation shows that ( n i= f H k,...,k n (x) = k ) i H k,...,k 2 n (x). Thus, each H k,...,k n is an eigenfunction of f, associated with the eigenvalue n i= λ k,...,k n = k i. 2 Since {H k,...,k n } is an orthogonal basis of L 2 (R n, γ n ), the real positive numbers λ k,...,k n will be all the possible eigenvalues, each one of then having finite multiplicity. Therefore, and By a direct calculation, we have σ ess ( f ) [, ) \ {i/2, i N}. Ric f = Ric R n + Hess f = 2, 2 f 2 = x 2 x = x 2 4 = f. Therefore, using Theorem.2, we have σ ess ( ) = [, + ). 3. WEYL S CRITERIA FOR THE ESSENTIAL SPECTRUM As tool for studying the essential spectrum, we will use a Weyl s Criteria. The following two criteria are corollaries of the classical one, and their proofs may be found in []. Proposition 3.. If there exists a sequence {u i } C (M) such that u i L. (M) ( λ)u i L (M) u i 2 as i ; L 2 (M) 2. for any compact K M, there exists i such that the support of u i is outside K for i > i ; and 3. (supp(u i )) is a C (n )-submanifold of M, then λ σ ess ( ). Sob a supervisão CPq/ICMC

6 48 LEONARDO SILVARES Proposition If there exists a sequence {u i } C (M) such that u i L (M) ( f λ)u i L f (M) u i 2 L 2 f (M) as i ; 2. for any compact K M, there exists i such that the support of u i is outside K for i > i ; and 3. (supp(u i )) is a C (n )-submanifold of M, then λ σ ess ( f ). These Propositions are key arguments to the proofs of Theorems.,.2 and.3. For each λ >, we will construct a sequence of functions satisfying conditions () to (3). The first attempt is to define each function u i in such way that u i (x) deppends only on the distance r(x) = d(x, p) of x to a fixed point p M. This approach, which seens very natural, fails because the distance r(x) may be not smooth and therefore u i (r(x)) may not belong to C (M), as required. The solution is to build an smooth approximation to the distance, as done in [9], [3] and [], and stated below. Proposition 3.3. Given p M, f C (M) and a decreasing continuous function δ : R + R +, such that lim r δ(r) =, there exists functions b and r in C (M) satisfying. b L f (M\B p(r)) δ(r); 2. r r L f (M\B r(p)) δ(r); 3. r(x) r(x) δ(r(x)) and r(x) 2, x M, r(x) > 2, and 4. f r(x) sup y B x() { f r(y)} + δ(r(x)) + b(x), x M, r(x) > 2 in the sense of distributions. Conditions, 2 and 4 say that the approximation is good in an average sense, and condition 3 gives a pontwise criterion of the approximation. If we define the r-balls B p (t) := {x M, r(x) < t}, given a >, the construction of r may be made [] so that, if vol f (M) =, and, if vol f (M) <, a vol f (B p (t)) vol f ( B p (t)) a vol f (B p (t)), [ a vol f (M) vol f ( B ] p (t)) vol f (M) vol f (B p (t)) a [vol f (M) vol f (B p (t))], (3.) Sob a supervisão da CPq/ICMC

7 ON THE ESSENTIAL SPECTRUM OF THE LAPLACIAN AND DRIFTED LAPLACIAN 49 for all t >. These comparisons between the volumes of r-balls and r-balls will be usefull to the calculations of the norms in condition of Propositions 3. and ESTIMATES FOR VOLUME AND LAPLACIANS OF DISTANCE Before we discuss the proofs of main Theorems, we present some results concerning the volume and the f-volume of B p (r). There are many classic results in Geometry which establishes some control on the growth and the decay of the volume, provided Ricci curvature has a lower bound. In this Section, we we obtain some similar results, using our hypotheses on the Bakry-Émery-Rici curvature tensor Ric f. If vol(m) >, we say that vol grows exponentially if there is C > such that vol(b p (r)) > e Cr, and, if vol f (M) <, we say that the volume of M decays exponentially at p if there exists C > such that vol(m) vol(b p (r)) < e Cr, for all r large enough. This definitions are naturally extended to vol f. In the scenario studied in Theorem., we have the following estimate, from [2]: Proposition 4.. Let M be a non-compact complete manifold. If f : M R is a f smooth function such that Ric f and lim r r =, then vol f does not grow nor decay exponentially. In the hypotheses of Theorem.2, Munteanu and Wang [] proved that the volume growth is at most Euclidean, and at least linear. In other words, fixed p M, there exists c, C R such that c r vol(b p (r)) C r n. In the hypotheses of Theorem.3, i.e. if the hypotheses of Theorem.2 are verified only in an end E of M, the volume growth remains at most Euclidean (see [2, Lemma 4.5], an adaptation of []), but there is no guarantee that is at least linear. We have, however, the following estimate: Proposition 4.2. If Ric f 2 g and f 2 f in some end E of M and vol(e) <, then vol(e \ B p (r)) does not decay exponentially. In order to prove the Proposition above, we need a deeper understanding of weight function f. The next Lemma, which is an adaptation of [, Proposition 4.2], shows that f has quadratic growth. Sob a supervisão CPq/ICMC

8 5 LEONARDO SILVARES Lemma 4.. In the Hypotheses of Theorem.3, given p M, there exists a constant c p > such that for any x E. 4 (d(p, x) c p) 2 f(x) 4 (d(p, x) + c p) 2, Proof: Let γ be a minimizing geodesic from p to a point of E. Since f 2 f, we have (2 f). (4.) Integrating along γ, we get f(t) 2 (t r ) + f(t). By taking c = sup y Sp(r ){ f(y) r 2 }, we have f(t) 2 t + c, which proves the upper bound. By the second variation formula of arc-length, along γ we have Ric(γ, γ )φ 2 (n ) (φ ) 2 γ for any φ with compact support on γ. Now set, r + t r t r φ =, r t r + r t, r t r, t > r. Using that Ric f = Ric +f /2, it follows that (n ) γ (φ ) 2 2 γ φ2 γ f (t)φ 2 (t)dt = 2 (r r ) + c + 2 γ f (t)φ(t)φ (t)dt, and so 2 (r r ) c 2 2 f (t)φ(t)φ (t)dt, (4.2) γ where c 2 = c 2 (n). By (4.), 2 f(s) f(t) s t, so r r r f (t)φ(t)dt Sob a supervisão da CPq/ICMC r f(t)φ(t)dt ( f(r) + /2) r γ r φ(t)dt = 2 f(r) + 4.

9 ON THE ESSENTIAL SPECTRUM OF THE LAPLACIAN AND DRIFTED LAPLACIAN 5 Hence, by (4.2), ( where c 3 is the supremum of 2 (r r ) c 2 2 γ f (t)φ(t)φ (t)dt = 2 r + r f (t)φ(t)dt + 2 r r f (t)φ(t)dt c r r f (t)φ(t)φ (t)dt c 3 + f(r) + 4, 2 r + r ) f (t)φ(t)dt on all geodesics γ as above. So f(r) 2 r c 4. Proof of Proposition 4.: Suppose by contradiction the existence of C > such that vol(e \ B p (r)) e Cr. (4.3) Now let R = r +, take q E such that d(p, q) = R + + S, for arbitrary S >, and denote J(t, ξ) dt dξ = d vol exp(tξ), the volume form in geodesic polar coordinates from q. Let p R be a point in the minimizing geodesic from p to q, and such that d(p R, p) = R. Taking a, < a <, let x be an arbitrary point of B q (S + a) B pr () and γ be the minimizing geodesic such that γ() = q and γ(s + a) = x. Notice that, since γ is a minimizing geodesic, if we define y := γ(2 + a), each x will determine, in an injective way, an unique y B q (2 + a). Moreover, we have y = exp q (a y ξ y ), where a y = a and ξ y = γ (). By Lemma 4., 4 (d(p, x) c p) 2 f(x) 4 (d(p, x) + c p) 2 Sob a supervisão CPq/ICMC

10 52 LEONARDO SILVARES for some c p R and for all x E. Denoting f(t) = f(γ(t)), for t T := S + a, and remembering γ() = q, γ(t ) = x we have f (t) f(t) 2 (d(γ(t), p) c p) 2 ((T t + R + ) + c p) = (T t + c), (4.4) 2 where c depends only on n, r and p. Similarly, adjusting c if necessary, By the Bochner formula, f(t) 4 (T t c)2 (4.5) f(t) 4 (T t + c)2 (4.6) ( t) (t) + n (f) = Ric( t, t) 2 + f (t). Multiplying by t k, k 2, integrating by parts and rearranging terms from to s < T, t(s) (n )k2 4(k )s 2(k + ) s + f (s) k s k Integrating from b := a + 2 to T, we have log(j(t, ξ)) log(j(b, ξ)) = b = b Some simple calculations lead to ( k s k s Sob a supervisão da CPq/ICMC b d log(j(s, ξ))) ds ds t(s) ds s (n )k2 4(k ) log T 4(k + ) T 2 + f (t)t k dt. ) f (t)t k dt ds = (f(t ) f(b)) + k ( k k t k b ( k s k s s ) f (t)t k dt ds. ) s=t f (t)t k dt s=b. (7)

11 ON THE ESSENTIAL SPECTRUM OF THE LAPLACIAN AND DRIFTED LAPLACIAN 53 Using (4.5) and (4.6), we have now (f(t ) f(b)) k k = for some c depending only on n, r and p. By (4.5), we have k k T k f (t)t k dt = ( ) 4 c2 (T b + c)2 4 ( (T b) 2 + 2(T b)c ) 4(k ) ( k 4 c2 ) (T b + c)2 4 = 4(k ) T 2 + c T, (8) k k k f(t ) c 2 k T k = c 2 k 4T k k(t c)2 4T k k T k f(t)t k 2 dt (T t c) 2 t k 2 dt t k dt + t k 2 dt 2k(T c) 4T k t k dt k = c 2 4(k + ) T 2 + k(t c)2 (T c)t 2 4(k ) = c 2 2(k 2 ) T 2 c + 2(k ) T (9) where c 2 = sup y E Bp(r +2){f(y)} depends only on n, r and p. Now using (4.4) we get k k b k b f (t)t k dt where c 3 depends only on n, r and p. Putting (7), (8) and (9) into (7), we get ( k s ) s k f (t)t k dt ds b k 2(k ) b k b (T t + c)t k dt = T + c 2(k ) b k 2(k )(k + ) b2 T = 2(k ) b + c 3 () = 4(k ) T 2 2(k 2 ) T 2 + c 4 k T 4(k + ) T 2 + c 4 T, () k Sob a supervisão CPq/ICMC

12 54 LEONARDO SILVARES where c 4 = c 4 (n, r, p). Replacing this in (4.6), we get log(j(t, ξ)) log(j(b, ξ)) = (n )k2 4(k ) log T If now we take in (), we get which implies 4(k+) T 2 + c 5 k log T c6 k T c 5 k log T c6 k T. T k = log T + log(j(t, ξ)) log(j(b, ξ)) c 7 T log T, J(b, ξ) e c7 T log T J(T, ξ). c4 4(k+) T 2 + c6 k T Since S T S + and a, there exists c 8 and c 9 constants and independent from a, ξ, S, q such that J(b, ξ) c 8 e c9 S log S J(S + a, ξ). Now, let U (B q (3)) and V T q M be, respectively, the set of all y and (a y, ξ y ) constructed as above. Notice that each x B q () determines an unique y in U and (a y, ξ y ) in V. Therefore e C(R+S 3) vol(m \ B p (R + S 3)) vol(b q (3)) vol(u) J(a, ξ) dr dξ V c 8 e c9 S log S J(S + a, ξ) dr dξ V c 8 e c9 S log S exp q (V ) d vol g c 8 e c9 S log S vol(b pr ()). Since M is complete, we take K = min pr vol(b pr ()) and so Therefore which is an absurd for large S. Sob a supervisão da CPq/ICMC ( e C(R 3)) e CS = e C(R+S 3) Kc 8 e c9 S log S. [ e c9 S log S Kc 8 e C(R 3) ] e CS,

13 ON THE ESSENTIAL SPECTRUM OF THE LAPLACIAN AND DRIFTED LAPLACIAN 55 Example 4.. If the end E is given by E = [2, + ) S 2, g E = dr 2 + r dθ2 and f E = r2 2, we have Ric f 2 and vol(e) <. This example shows that, unlike what happens in Theorem.2, in the hypotheses of Theorem.3 the volume of E may be finite. In addition to estimates of the growth and decay volume, we need some estimates on r and r to prove our main Theorems. Lemma 4.2. In the hypotheses of Theorem., lim r f r, and, in the hypotheses of Theorems.2 or.3, lim r r. This Lemma gives us the following three Lemmas. Lemma 4.3. t > R, For all ε > and t large enough, there exists R = R(ε, t ) such that, for in the hypotheses of Theorem. and assuming vol f (M) =, f r e f d vol g ε vol f (B p (t + )) ; B p(t)\b p(t ) in the hypotheses of Theorem.2 (remember that vol(m) = ), r d vol g ε vol(b p (t + )) ; B p(t)\b p(t ) in the hypotheses of Theorem.3 and assuming vol(m) =, r d vol g ε vol (E B p (t + )). (B p(t)\b p(t )) E Lemma 4.4. Given ε >, the construction of r in Proposition 3.3 may be made so that, for R > large enough, we have, for all t > R, in the hypotheses of Theorem. and assuming vol f (M) <, f r e f d vol g ε(vol f (M) vol f (B p (t))) + 2 vol f ( B p (t)) ; M\B p(t) in the hypotheses of Theorem.3 and assuming vol(m) <, r d vol g ε(vol(e \ B p (t))) + 2 vol( B p (t)). E\B p(t) Sob a supervisão CPq/ICMC

14 56 LEONARDO SILVARES for t > R. Lemma 4.5. Let M and f be as in the hypotheses of Theorem.3 and suppose vol(m) <. Given ɛ >, C >, there exist R > large and a sequence of real numbers r k such that ɛ[vol(m) vol(b p (r k R))] + C vol( B(r k R)) 2ɛ [vol(m) vol(b(r k ))]. The four previous Lemmas are proved in [2] and [] (hypotheses of Theorems. and.2 only). 5. PROOF OF THEOREM.3 According to Proposition 3., in order to prove that σ ess ( ) = [, ) in L 2, we need only to construct, for all λ >, a sequence {u i } C (M) satisfying. u i L (M) ( λ)u i L (M) as i ; u i L 2 (M) 2. u i weakly; and 3. (supp(u i )) is a C (n )-submanifold of M. The proof will be divided in two cases, vol(e) = and vol(e) <. For the sake of simplicity ( of the notation, we will write V (r) instead of vol(e B p (r)), and Ṽ (r) instead of vol E B ) p (r). Proof of Theorem.3, case vol(e) = : Given p M, let r be the C approximation for r we have constructed in Proposition 3.3. Let x, y, R be such that < R < x < y and whose values will be chosen later. Let ψ : R R be a cut-off function satisfying ψ(r) = so that ψ and ψ are bounded. Given λ >, we define Sob a supervisão da CPq/ICMC {, r [x/r, y/r], r / [x/r, y/r + ] ( ) r(x) φ(x) = ψ e i λ r. R

15 ON THE ESSENTIAL SPECTRUM OF THE LAPLACIAN AND DRIFTED LAPLACIAN 57 Hence, φ and So we have and therefore, φ + λφ = [ ( ) ψ R R + 2i λψ ( λ r 2). r 2 + φ + λφ C + C r + C r r R (i ) ] λψ + ψ R r e i λ r φ + λφ L (M) C (Ṽ (y + R) Ṽ (x R)) + Cδ(x R) R +C r d vol g. () (B p(y+r)\b p(x R)) E By Lemma 4.3 we choose y large enough such that (B p(y+r)\b p(x R)) E r d vol g εv (y + R + ). (2) On the other hand, fixing x and R with R and x R large enough, we have C (Ṽ (y + R) Ṽ (x R)) < ε(ṽ (y + R) Ṽ (x R)) R < εṽ (y + R + ) (in the last inequality above we have inequality (3.) ) and Using (2), (3) and (4) in (), we get < 2εV (y + R + ) (3) Cδ(x R) < εv (y + R + ). (4) φ + λφ L (M) < 4εV (y + R + ). Since φ in B p (y) \ B p (x), we have φ 2 L 2 (M) Ṽ (y) Ṽ (x). Fixing x and taking y large enough, φ 2 L 2 (M) 2Ṽ (y) and, by (3.), φ 2 L 2 (M) 4V (y). As a consequence of Proposition 4.2, taking y even larger, we have V (y + R + ) 2V (y), so, φ L (M) ( λ)φ L (M) < φ L 2 (M) 4εV (y + R + ) 4 V (y) 8εV (y) < 32ε. 4V (y) Sob a supervisão CPq/ICMC

16 58 LEONARDO SILVARES Thus, since φ L (M) =, we have constructed a compact supported function φ such that φ L (M) ( λ)φ L (M) < 32ε φ L 2 (M) for an arbitrarily small ε >. In order to obtain the desired sequence u n, we consider, in the above construction, ε = /n and u n = φ, with x greater than the o value of y + R of the construction of u n. Proof of Theorem.3, case vol(e) < : In the construction of r in Proposition 3.3, we take δ such that δ(r) (vol(m) V (r)). (5) r By Lemma 4.4, we choose x large enough so that (B p(x+r)\b p(x R)) E r d vol g E\B p(x R) r d vol g ε(vol(m) V (x R)) +2 vol( B p (x R)). By using this in () of case vol(e) =, we have φ + λφ L (M) C R (Ṽ (y + R) Ṽ (x R)) + Cδ(x R)+ +Cε(vol(M) V (x R)) + 2C vol( B p (x R)) C R (vol(m) Ṽ (x R)) + Cδ(x R)+ +Cε(vol(M) V (x R)) + 2C vol( B p (x R)) 2 C R (vol(m) V (x R)) + Cδ(x R)+ +Cε(vol(M) V (x R)) + 2C vol( B p (x R)) C ( 2 R + ε) (vol(m) V (x R)) + Cδ(x R)+ +2C vol( B p (x R)) (in the third inequality above we have used inequality (3.) ). Since M has finite volume, and by (5), we may choose R and x so that and Thus 2 ε, R δ(x R) ε(vol(m) V (x R)). φ + λφ L (M) 3Cε(vol(M) V (x R)) + 2C vol( B p(x R)). Sob a supervisão da CPq/ICMC

17 ON THE ESSENTIAL SPECTRUM OF THE LAPLACIAN AND DRIFTED LAPLACIAN 59 By Lemma 4.5, taking x even larger, therefore, 3ε(vol(M) V (x R)) + 2 vol( B p (x R)) 6ε (vol(m) V (x)), φ + λφ L (M) 6ε C(vol(M) V (x)). (6) Using again that the volume of M is finite, making y large enough, we get and, by (6), we have Ṽ (y) Ṽ (x) 2 (vol(m) Ṽ (x)) (vol(m) V (x)), 4 φ + λφ L (M) 6ε C(vol(M) V (x)) 24ε C(V (y) V (x)) 24ε C φ 2 L 2 (M). The proof follows now the case of infinite volume, as above. The proof of Theorem. is analogous to the proof of Theorem.3, since all Lemmas and Propositions used in the proof above are equally valid in the hypotheses of Theorem.. The same may be applied to Theorem.2 and, even it is a particular case of Theorem.3, an independent proof could also be obtained. REFERENCES. D. Bakry and M. Émery. Diffusions hypercontractives. Lecture Notes in Math., 23:77 26, Springer, H.-D. Cao and D. Zhou. On complete gradient shrinking Ricci solitons. J. Differential Geom., 85(2):75 86, N. Charalambous and Z. Lu. The essential spectrum of the Laplacian. Available at arxiv:2.3225, Z. H. Chen and Z. Q. Lu. Essential spectrum of complete Riemannian manifolds. Sci. China Ser. A, 35(3):477 5, H. Donnelly. Exhaustion functions and the spectrum of a complete Riemannian manifold. Indiana Univ. Math. J., 46(2):55 527, J. F. Escobar. On the spectrum of the Laplacian on complete Riemannian manifolds. Comm. Partial Differential Equations, 65():63 85, J. F. Escobar and A. Freire. The spectrum of the Laplacian of manifolds of positive curvature. Duke Math. J., 65(): 2, J. Y. Li. Spectrum of the Laplacian on a complete Riemannian manifolds with nonnegative Ricci curvature which possess a pole. J. Math. Soc. Japan, 46(2):23 26, Z. Lu and D. Zhou. On the essential spectrum of complete non-compact manifolds. J. Funct. Anal., 26(): , 2. Sob a supervisão CPq/ICMC

18 6 LEONARDO SILVARES. O. Munteanu and J. Wang. Geometry of manifolds with densities. Advances in Mathematics, 259(): , 24.. L. Silvares. On the essential spectrum of the Laplacian and the drifted Laplacian. J. Funct. anal., 266: , L. Silvares. On the essential spectrum of the Laplacian and the drifted Laplacian. PhD thesis, Universidade Federal Fluminense, April K-T. Sturm. On the L p -spectrum of uniformly elliptic operators on Riemannian manifolds. J. Funct. anal., 8: , J. Wang. The spectrum of the Laplacian on a manifold of nonnegative Ricci curvature. Math. Res. Lett., 4(4): , G. Wei and W. Wylie. Comparison geometry for the Bakry-Émery tensor. J. Differential Geom., 2(83):337 45, D. Zhou. Essential spectrum of the Laplacian on manifolds of nonnegative curvature. Int. Math. Res. Not., 5, 994. Sob a supervisão da CPq/ICMC