Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons
aa r X i v : . [ m a t h . DG ] J un SHARP GAUSSIAN UPPER BOUNDS FOR SCHR ¨ODINGER HEATKERNEL ON GRADIENT SHRINKING RICCI SOLITONS
JIA-YONG WU
Abstract.
For gradient shrinking Ricci solitons, we observe that the study of Schr¨odingerheat kernel seems to be more natural than the classical heat kernel. In this paper wederive sharp Gaussian upper bounds for the Schr¨odinger heat kernel on complete gradientshrinking Ricci solitons with the scalar curvature bounded above. As applications, we provesharp upper bounds for the Green’s function of the Schr¨odinger operator. We also provesharp lower bounds for eigenvalues of the Schr¨odinger operator. These sharp cases all takeplace at Euclidean Gaussian shrinking Ricci solitons. Introduction
In this paper we will investigate Gaussian upper estimates for Schr¨odinger heat kernelson complete gradient shrinking Ricci solitons and their applications. Let (
M, g ) be an n -dimensional complete Riemannian manifold and let f be a smooth function on ( M, g ). Thetriple (
M, g, f ) is called a complete gradient shrinking Ricci soliton (see [16]) ifRic + Hess f = 12 g, (1.1)where Ric is the Ricci curvature of ( M, g ) and Hess f is the Hessian of f . The function f is called a potential function for the gradient Ricci soliton. Gradient Ricci solitons play animportant role in the Ricci flow [16] and Perelman’s [25, 26, 27] resolution of the Poincar´econjecture and the geometrization conjecture. See [2] for an excellent survey.For a gradient shrinking Ricci soliton, we consider Schr¨odinger heat kernels of the operator L = − ∆ + a R , where ∆ is the Laplace operator with respect to g , R is the scalar curvature of ( M, g, f )and a > H R ( x, y, t ), definedon M × M × R + , denote the Schr¨odinger fundamental solution of the operator L . Thatis, for each y ∈ M , H R ( x, y, t ) = u ( x, t ) is a smooth solution of the Schr¨odinger-type heatequation ∆ u − ∂ t u − a R u = 0satisfying the initial condition lim t → u ( x, t ) = δ y ( x ), where δ y ( x ) is the delta functiondefined by Z M φ ( x ) δ y ( x ) dv = φ ( y ) Date : June 25, 2020.2010
Mathematics Subject Classification.
Primary 35K08; Secondary 53C21, 58J50.
Key words and phrases. gradient shrinking Ricci soliton, heat kernel, Schr¨odinger operator, Green’sfunction, eigenvalue. for any φ ∈ C ∞ ( M ). The Schr¨odinger heat kernel for the operator L is defined to bethe minimal positive Schr¨odinger fundamental solution. We will see that, when the scalarcurvature of ( M, g, f ) is bounded from above by a constant, the Schr¨odinger heat kernelalways exists on gradient shrinking Ricci solitons (for the explanation, see Section 2).Several factors motivate us to study the Schr¨odinger operator L := − ∆ + a R instead ofthe classical Laplace operator − ∆ on gradient shrinking Ricci solitons. First, the Perel-man’s geometric operator − ∆ + R [25] was widely considered in the Ricci flow theory.Regarding that shrinking Ricci solitons are the self-similar solutions to the Ricci flow [16],the Schr¨odinger operator L and the Perelman’s geometric operator, seem to be more naturalcompared with the Laplace operator for the gradient Ricci solitons. Second, the gradientshrinking Ricci soliton is more or less related to the Yamabe invariant [1, 25], which leads tothe Schr¨odinger operator L is similar to the conformal Laplacian − ∆ + n − n − R. Third, forgradient shrinking Ricci solitons, Li and Wang [21] proved a Sobolev inequality including ascalar curvature term, which inspires us to consider the Schr¨odinger operator L instead ofthe Laplace operator.On gradient shrinking Ricci soliton ( M, g, f ), according to a nice observation of Carrilloand Ni (Theorem 1.1 in [5]), by adding a constant to f , without loss of generality, we mayassume (see the explanation in Section 2 or [21])(1.2) R + |∇ f | = f and Z M (4 π ) − n e − f dv = e µ , where R is the scalar curvature of ( M, g ) and µ = µ ( g,
1) is the entropy functional ofPerelman [25]. For the Ricci flow, the Perelman’s entropy functional is time-dependent, buton a fixed gradient shrinking Ricci soliton it is constant and finite.The following Gaussian upper bound for the Schr¨odinger heat kernel H R ( x, y, t ) is themain result of this paper, similar to the Gaussian heat kernel estimate for Laplace operatoron Riemannian manifolds. This result will be useful for understanding the function theoryof gradient shrinking Ricci solitons. Theorem 1.1.
Let ( M, g, f ) be an n -dimensional complete gradient shrinking Ricci solitonsatisfying (1.1) and (1.2) with scalar curvature R bounded from above by a constant. For any c > , there exists a constant A = A ( n, c ) depending on n and c such that the Schr¨odingerheat kernel of the operator − ∆ + a R with a ≥ satisfies (1.3) H R ( x, y, t ) ≤ Ae − µ (4 πt ) n exp (cid:18) − d ( x, y ) ct (cid:19) for all x, y ∈ M and t > , where µ is the Perelman’s entropy functional.Remark . The upper assumption on scalar curvature only guarantees the Schr¨odingerheat kernel admits similar propositions of the classical heat kernel of the Laplace operator,such as existence, semigroup property, eigenfunction expansion, etc (see Section 2). It seemsnot be directly used in the proof of Theorem 1.1. It is interesting to ask if the Schr¨odingerheat kernel still exists on complete noncompact gradient shrinking Ricci solitons when thescalar curvature assumption is removed.
HARP GAUSSIAN UPPER BOUNDS FOR SCHR ¨ODINGER HEAT KERNEL 3
Remark . The classical heat kernel of the Laplace operator on an n -dimensional Euclideanspace R n is H ( x, y, t ) = 1(4 πt ) n exp (cid:18) − | x − y | t (cid:19) for all x, y ∈ R n and t >
0. This indicates the Gaussian upper estimate (1.3) is sharp (atthis time R = µ = 0).Recall that for the heat kernel of the Laplace operator, Cheng, Li and Yau [8] provedupper Gaussian estimates on manifolds satisfying bounded sectional curvature and a lowerbound of the injectivity radius, which was later generalized by Cheeger, Gromov and Taylor[7] to manifolds with the Ricci curvature bounded below. In 1986, Li and Yau [20] used thegradient estimate technique to derive sharp Gaussian upper and lower bounds on manifoldswith nonnegative Ricci curvature. In 1990s, Grigor’yan [13] and Saloff-Coste [28] indepen-dently proved similar estimates on manifolds satisfying the volume doubling property andthe Poincar´e inequality, by using the Moser iteration technique. Davies [10] further devel-oped Gaussian upper bounds under a mean value property assumption. Recently, the firstauthor and P. Wu [30] applied De Giorgi-Nash-Moser theory to derive sharp Gaussian upperand lower bound estimates for the weighted heat kernel on smooth metric measure spaceswith nonnegative Bakry- ´Emery Ricci curvature. For the heat kernel of a general Schr¨odingeroperator − ∆ + V for some V ∈ C ∞ ( M ), many authors studied global bounds for the heatkernel on manifolds. For example, see [9, 11, 12, 15, 20, 23, 29, 33, 34, 35, 36] and referencestherein.The proof strategy for Theorem 1.1 seems to be different from the above mentionedmethods, and here its proof mainly includes two steps. The first step is that we applya local Logarithmic Sobolev inequality of shrinking Ricci solitons to give an upper boundof the Schr¨odinger heat kernel (see Theorem 3.1), which is motivated by the argument ofmanifolds [9]. The second step is that we extend the upper bound of Schr¨odinger heat kernelto a delicate upper bound with a Gaussian exponential factor, whose argument involvingupper estimates for a weighted integral of Schr¨odinger heat kernel (see Proposition 4.1), byusing a delicate iteration technique due to Grigor’yan [14].Now we give two applications of the Schr¨odinger heat kernel estimates. On one handwe derive upper bounds for the Green’s function of the Schr¨odinger operator on gradientshrinking Ricci solitons, similar to the classical estimates of Li and Yau [20] for manifolds.Recall that, for a complete gradient shrinking Ricci soliton ( M, g, f ), the Green’s functionof the Schr¨odinger operator − ∆ + a R with a ≥ is defined by G R ( x, y ) := Z ∞ H R ( x, y, t ) dt if the integral on the right hand side converges. Theorem 1.4.
Let ( M, g, f ) be an n -dimensional ( n ≥ complete gradient shrinking Riccisoliton satisfying (1.1) and (1.2) with scalar curvature R bounded above by a constant. If G R ( x, y ) exists, then for any c > , there exists a constants B ( n, c ) depending on n and c ,such that (1.4) G R ( x, y ) ≤ B ( n, c ) e − µ r n − , JIA-YONG WU where r = r ( x, y ) and µ is the Perelman’s entropy functional.Remark . The exponent n − R n , g E , | x | ), where g E is the standard flat Euclidean metric, we haveR = 0 and µ = 0. At this time G R ( x, y ) is just the classical Euclidean Green’s functiongiven by G R ( x, y ) = C ( n ) r ( x, y ) − n for some positive constant C ( m ), where n ≥ L on compact gradient shrinking Ricci solitons,by adapting the argument of Laplace operator on manifolds [20]. Theorem 1.6.
Let ( M, g, f ) be an n -dimensional closed gradient shrinking Ricci solitonsatisfying (1.1) and (1.2) . Let { < λ ≤ λ ≤ . . . } be the set of eigenvalues of theSchr¨odinger operator − ∆ + a R with a ≥ . Then λ k ≥ nπe (cid:18) k e µ V ( M ) (cid:19) /n for all k ≥ , where V ( M ) is the volume of M and µ is the Perelman’s entropy functional.Remark . From the proof of Theorem 1.6 in Section 6, we can apply the same method toobtain similar eigenvalue estimates to allow the compact gradient shrinking soliton to haveconvex boundaries with either Dirichlet or Neumann boundary conditions.
Remark . For bounded domain Ω ⊂ R n , the well-known Weyl’s asymptotic formula ofthe k -th Dirichlet eigenvalue of the Laplace operator satisfies λ k (Ω) ∼ c ( n ) (cid:18) kV (Ω) (cid:19) /n , k → ∞ , where c ( n ) is the Weyl constant with c ( n ) = 4 π ω − /nn , ω n is the volume of the unit ball in R n , which indicates our lower eigenvalue estimates are sharp for the exponent 2 /n (at thistime R = µ = 0). Moreover the constant nπe ≤ c ( n ) and has the asymptotic propertylim n →∞ nπe · c ( n ) = 1 . We remark that Li and Yau [19] used the Fourier transform method to get lower boundsfor Dirichlet eigenvalues of the Laplace operator on a bounded domain Ω ⊂ R n , which waslater generalized by them [20] to the manifolds with the Ricci curvature bounded below.Grigor’yan [15] proved lower bounds for eigenvalues of the Schr¨odinger operator under someassumption on the first eigenvalue. The author and P. Wu [30] obtained lower estimates foreigenvalues of the weighted Laplace operator on compact weighted manifolds.The paper is organized as follows. In Section 2, we recall some basic properties ofSchr¨odinger heat kernels. We also introduce some identities and a Logarithmic Sobolevinequality [21] on gradient shrinking Ricci solitons. In Section 3, we apply the Logarithmic HARP GAUSSIAN UPPER BOUNDS FOR SCHR ¨ODINGER HEAT KERNEL 5
Sobolev inequality to prove the ultracontractivity of Schr¨odinger heat kernels. In Section 4,we shall prove Theorem 1.1. Precisely, we use the ultracontractivity to obtain the Gaussiantype upper bounds on shrinking Ricci solitons. In Section 5, we apply Theorem 1.1 to de-rive an upper bound of the Green’s function for the Schr¨odinger operator. In Section 6, forcompact gradient shrinking Ricci solitons, we apply upper bounds of the Schr¨odinger heatkernel to give eigenvalue estimates of the Schr¨odinger operator.
Acknowledgement . The author thanks Professor Qi S. Zhang for helpful discussions.This work was partially supported by NSFS (17ZR1412800) and NSFC (11671141).2.
Preliminaries
In this section, we summarize some basic facts about Schr¨odinger heat kernels and gradientshrinking Ricci solitons. First we present some basic results regarding the Schr¨odinger heatkernel on manifolds. Recall that, according to Theorem 24.40 of [3], we have the existenceof the Schr¨odinger heat kernel on Riemannian manifolds.
Theorem 2.1.
Let ( M, g ) be a complete Riemannian manifold. If smooth function Q ( x ) onmanifold ( M, g ) is bounded, then there exists a unique smooth minimal positive fundamentalsolution H Q ( x, y, t ) for the heat-type operator ∂ t − ∆ + Q . As in the classical case, the Schr¨odinger heat kernel H Q ( x, y, t ) can be regarded as thelimit of the Dirichlet Schr¨odinger heat kernels on a sequence of exhausting subsets in M ;see [3]. The idea of the proof is as follows. Let Ω ⊂ Ω ⊂ ... ⊂ M be an exhaustion ofrelatively compact domains with smooth boundary in ( M, g ). In each Ω k , k = 1 , ... , wecan construct the Dirichlet heat kernel H Q Ω k ( x, y, t ) for the heat-type operator ∂ t − ∆ + Q .By the maximum principle we have 0 ≤ H Q Ω k ≤ H Q Ω k +1 , and Z Ω k H Q Ω k ( x, y, t ) dv ( x ) ≤ . These properties assures H Q ( x, y, t ) = lim k →∞ H Q Ω k ( x, y, t ) . Moreover, the Schr¨odinger heat kernel satisfies the symmetry property H Q Ω k ( x, y, t ) = H Q Ω k ( y, x, t ) , H Q ( x, y, t ) = H Q ( y, x, t )and the semigroup identity H RΩ k ( x, y, t + s ) = Z Ω k H RΩ k ( x, z, t ) H RΩ k ( z, y, s ) dv ( z ) ,H R ( x, y, t + s ) = Z M H R ( x, z, t ) H R ( z, y, s ) dv ( z ) . Since Q is bounded, the Schr¨odinger operator L = − ∆ + Q is self-adjoint operator andthe spectrum of L shares similar properties with the Laplace operator case (see [12]). For acompact subdomain Ω ⊂ M , by the elliptic theory we let { ϕ k } ∞ k =0 in L (Ω) be the completeorthonormal sequence of the Dirichlet eigenfunctions of the Schr¨odinger operator L with the JIA-YONG WU corresponding nondecreasing sequence of discrete eigenvalues { λ k } ∞ k =1 satisfying { < λ ≤ λ ≤ . . . } . Then the Dirichlet Schr¨odinger heat kernel of L has the eigenfunction expansion H Q Ω ( x, y, t ) = ∞ X k =1 e − λ k t ϕ k ( x ) ϕ k ( y ) . Clearly, this expansion can be used to closed manifold M , i.e., Ω = M .In particular for this paper, on complete gradient shrinking Ricci soliton, we consider heatkernels of the Schr¨odinger operator L = − ∆ + a R, where a > R is bounded from above by a constant, by Theorem 2.1, then the Schr¨odinger heat kernel H R ( x, y, t ) always uniquely exists on complete gradient shrinking Ricci soliton ( M, g, f ).Meanwhile, the above mentioned propositions all hold for H R ( x, y, t ).Next we explain why (1.2) holds on gradient shrinking Ricci soliton (1.1). Using (1.1),we get R + ∆ f = n , and(2.1) C ( g ) := R + |∇ f | − ( f + c )is a finite constant, where c ∈ R is a free parameter (see Chapter 27 in [4]). Combiningthese equalities gives(2.2) 2∆ f − |∇ f | + R + ( f + c ) − n = − C ( g ) . On an n -dimensional complete Riemannian manifold ( M, g ), the Perelman’s W -entropyfunctional [25] is defined by W ( g, φ, τ ) := Z M h τ (cid:16) |∇ φ | + R (cid:17) + φ − n i (4 πτ ) − n/ e − φ dv for some φ ∈ C ∞ ( M ) and τ >
0, when this entropy functional is finite, and the Perelman’s µ -entropy functional [25] is defined by µ ( g, τ ) := inf n W ( g, φ, τ ) (cid:12)(cid:12)(cid:12) φ ∈ C ∞ ( M ) with Z M (4 πτ ) − n/ e − φ dv = 1 o . Carrillo and Ni [5] gave an interesting observation that function f + c is always a minimizerof µ ( g,
1) on complete (possible non-compact) gradient shrinking Ricci soliton (
M, g, f ).Therefore, by (2.2), we have µ ( g,
1) = W ( g, f + c,
1) := Z M (cid:16) |∇ f | + R + ( f + c ) − n (cid:17) (4 π ) − n/ e − ( f + c ) dv = Z M (cid:16) f − |∇ f | + R + ( f + c ) − n (cid:17) (4 π ) − n/ e − ( f + c ) dv = − C ( g ) , where c is a constant such that R M (4 π ) − n/ e − ( f + c ) dv = 1. Notice that the above integralformulas always hold (see [17] for the detail explanation). HenceR + |∇ f | − ( f + c ) = − µ ( g, HARP GAUSSIAN UPPER BOUNDS FOR SCHR ¨ODINGER HEAT KERNEL 7 with R M (4 π ) − n/ e − ( f + c ) dv = 1. Letting c = µ ( g, Lemma 2.2.
Let ( M, g, f ) be an n -dimensional complete gradient shrinking Ricci solitonsatisfying (1.1) and (1.2) . For each compactly supported locally Lipschitz function ϕ with R Ω ϕ dv = 1 and each number τ > , Z Ω ϕ ln ϕ dv ≤ τ Z Ω (cid:0) |∇ ϕ | + R ϕ (cid:1) dv − h µ + n + n πτ ) i , where R is the scalar curvature of ( M, g, f ) and µ is the Perelman’s entropy functional. If the scalar curvature is bounded from above by a constant, Lemma 2.2 reduces tothe classical Logarithmic Sobolev inequality on manifolds. Li and Wang [21] used theLogarithmic Sobolev inequality to prove a Sobolev inequality on complete gradient shrinkingRicci solitons. In this paper, we will apply the Logarithmic Sobolev inequality to derivesharp Gaussian upper bounds of the Schr¨odinger heat kernel.3.
Ultracontractivity
In this section, we will apply the Logarithmic Sobolev inequality (Lemma 2.2) to de-rive upper bounds for the Schr¨odinger heat kernels on complete gradient shrinking Riccisolitons. In other words, we will show that the Schr¨odinger heat kernels possesses the ul-tracontractivity. A similar result has been explored for heat kernels of the Laplace operatoron Riemannian manifolds [9, 37].
Theorem 3.1.
Let ( M, g, f ) be an n -dimensional complete gradient shrinking Ricci solitonsatisfying (1.1) and (1.2) with scalar curvature R bounded above by a constant. Then theSchr¨odinger heat kernel of the operator − ∆ + a R with a ≥ satisfies (3.1) H R ( x, y, t ) ≤ e − µ (4 πt ) n for all x, y ∈ M and t > , where µ is the Perelman’s entropy functional.Remark . As in Remark 1.3, the assumption on scalar curvature only guarantees theexistence and semigroup property of the Schr¨odinger heat kernel H R ( x, y, t ) on completeshrinking Ricci solitons and it is not directly used in the proof of Theorem 3.1. Remark . Recently, Li and Wang [21] proved a similar upper bound of the conjugateheat kernel on ancient solutions of the Ricci flow induced by a Ricci shrinking soliton. Fora fixed-metric Ricci shrinking soliton, the study of the Schr¨odinger heat kernel seems to bemore reasonable than the evolved-metric setting.
Proof of Theorem 3.1.
By the approximation argument, it suffices to prove (3.1) for theDirichlet Schr¨odinger heat kernel H RΩ ( x, y, t ) of any compact set Ω in ( M, g, f ). In fact,let Ω i , i = 1 , , ... , be a sequence of compact exhaustion of M such that Ω i ⊂ Ω i +1 and JIA-YONG WU ∪ i Ω i = M . If we are able to prove (3.1) for the Dirichlet Schr¨odinger heat kernel H RΩ i ( x, y, t )for any i , then the result follows by letting i → ∞ .In the rest, we will use the argument of [9] (see also [37]) to give the estimate (3.1). Let u = u ( x, t ), t ∈ [0 , T ], be a smooth solution to the heat-type Schr¨odinger equation( ∂ t − L ) u = 0 , where L = − ∆+ a R, in a compact set Ω ⊂ M with Dirichlet boundary condition: u ( x, t ) = 0on ∂ Ω. Then u ( x, t ) can be written as u ( x, t ) = Z Ω u ( y, H RΩ ( x, y, t ) dv ( y ) , where H RΩ ( x, y, t ) denotes the Schr¨odinger heat kernel of the operator L in a compact setΩ ⊂ M .In the following, we shall estimate k u k p ( t ) := (cid:18)Z Ω | u | p ( t ) dv (cid:19) p ( t ) , where p ( t ) = TT − t , t ∈ [0 , T ], which obviously satisfies p (0) = 1 and p ( T ) = ∞ . To achieveit, we compute that ∂ t k u k p ( t ) = − p ′ ( t ) p ( t ) k u k p ( t ) · ln (cid:16) k u k p ( t ) p ( t ) (cid:17) + k u k − p ( t ) p ( t ) p ( t ) (cid:20) p ′ ( t ) Z Ω u p ( t ) ln udv + p ( t ) Z Ω u p ( t ) − u t dv (cid:21) = − p ′ ( t ) p ( t ) k u k p ( t ) · ln (cid:16) k u k p ( t ) p ( t ) (cid:17) + k u k − p ( t ) p ( t ) p ( t ) (cid:20) p ′ ( t ) Z Ω u p ( t ) ln udv + p ( t ) Z Ω u p ( t ) − ∆ udv − ap ( t ) Z Ω R udv (cid:21) . Multiplying function p ( t ) k u k p ( t ) p ( t ) in the above equality and integrating by parts for the term∆ u , we have p ( t ) k u k p ( t ) p ( t ) · ∂ t k u k p ( t ) = − p ′ ( t ) k u k p ( t ) p ( t ) · ln (cid:16) k u k p ( t ) p ( t ) (cid:17) + p ( t ) p ′ ( t ) k u k p ( t ) Z Ω u p ln udv − p ( t )( p ( t ) − Z Ω u p ( t ) − |∇ u | dv − ap ( t ) k u k p ( t ) Z Ω R u p dv. Dividing by k u k p ( t ) in the above equality yields p ( t ) k u k p ( t ) p ( t ) · ∂ t (cid:0) ln k u k p ( t ) (cid:1) = − p ′ ( t ) k u k p ( t ) p ( t ) · ln (cid:16) k u k p ( t ) p ( t ) (cid:17) + p ( t ) p ′ ( t ) Z Ω u p ln udv − p ( t ) − Z Ω |∇ u p ( t )2 | dv − ap ( t ) Z Ω R u p dv, HARP GAUSSIAN UPPER BOUNDS FOR SCHR ¨ODINGER HEAT KERNEL 9 which further implies(3.2) p ( t ) · ∂ t (cid:0) ln k u k p ( t ) (cid:1) = − p ′ ( t ) · ln (cid:16) k u k p ( t ) p ( t ) (cid:17) + p ( t ) p ′ ( t ) k u k p ( t ) p ( t ) Z Ω u p ln udv − p ( t ) − k u k p ( t ) p ( t ) Z Ω |∇ u p ( t )2 | dv − ap ( t ) k u k p ( t ) p ( t ) Z Ω R u p dv. We now introduce a new quantity to simplify equality (3.2). Set v := u p ( t )2 k u p ( t )2 k . Then, we see that v = u p ( t ) k u k p ( t ) p ( t ) , k v k = 1 and ln v = ln u p ( t ) − ln (cid:16) k u k p ( t ) p ( t ) (cid:17) . So, we have the equality p ′ ( t ) Z Ω v ln v dv = p ′ ( t ) Z Ω u p ( t ) k u k p ( t ) p ( t ) h ln u p ( t ) − ln (cid:16) k u k p ( t ) p ( t ) (cid:17)i dv = p ( t ) p ′ ( t ) k u k p ( t ) p ( t ) Z Ω u p ( t ) ln udv − p ′ ( t ) ln (cid:16) k u k p ( t ) p ( t ) (cid:17) . Using the above equality, (3.2) can be simplified as(3.3) p ( t ) ∂ t (cid:0) ln k u k p ( t ) (cid:1) = p ′ ( t ) Z Ω v ln v dv − p ( t ) − Z Ω |∇ v | dv − ap ( t ) Z Ω R v dv = p ′ ( t ) (cid:20)Z Ω v ln v dv − p ( t ) − p ′ ( t ) Z Ω |∇ v | dv − ap ( t ) p ′ ( t ) Z Ω R v dv (cid:21) . Now we want to apply the Logarithmic Sobolev inequality (Lemma 2.2) to estimate (3.3).Indeed in Lemma 2.2, we can choose ϕ = v and 4 τ = 4( p ( t ) − p ′ ( t ) = 4 t ( T − t ) T ≤ T. and Lemma 2.2 gives Z Ω v ln v dv ≤ ( p ( t ) − p ′ ( t ) Z Ω (cid:0) |∇ v | + R v (cid:1) dv − h µ + n + n πτ ) i . Using this, (3.3) can be reduced to p ( t ) ∂ t (cid:0) ln k u k p ( t ) (cid:1) ≤ p ′ ( t ) (cid:20) p ( t ) − − ap ( t ) p ′ ( t ) Z Ω R v dv − µ − n − n πτ ) (cid:21) . Since p ( t ) − − ap ( t ) = − a (cid:18) p ( t ) − a (cid:19) + (cid:18) a − (cid:19) ≤ , where we used a ≥ in the second inequality above, then p ( t ) ∂ t (cid:0) ln k u k p ( t ) (cid:1) ≤ p ′ ( t ) h − µ − n − n πτ ) i . Noticing that p ′ ( t ) p ( t ) = 1 T and τ = t ( T − t ) T , then we obtain ∂ t (cid:0) ln k u k p ( t ) (cid:1) ≤ T (cid:20) − µ − n − n πt ( T − t ) T (cid:21) . Integrating the above inequality from 0 to T with respect to t , we haveln (cid:18) k u ( x, T ) k p ( T ) k u ( x, k p (0) (cid:19) ≤ − µ − n π ) − n T. Notice that p (0) = 1 and p ( T ) = ∞ , and we have k u ( x, T ) k ∞ ≤ k u ( x, k · e − µ (4 πT ) n . Since u ( x, T ) = Z Ω u ( y, H RΩ ( x, y, T ) dv ( y ) , then we conclude H RΩ ( x, y, T ) ≤ e − µ (4 πT ) n and the result follows since T is arbitrary. (cid:3) By a similar argument, it is easy to see that when a = 0, we have Proposition 3.4.
Let ( M, g, f ) be an n -dimensional complete gradient shrinking Ricci soli-ton satisfying (1.1) and (1.2) . If the scalar curvature R of ( M, g, f ) satisfies R ≤ C R for some constant C R ≥ , then the heat kernel H ( x, y, t ) of the Laplace operator satisfies (3.4) H ( x, y, t ) ≤ e − µ (4 πt ) n exp (cid:18) C R t (cid:19) for all x, y ∈ M and t > , where µ is the Perelman’s entropy functional. In the end of this section, by using the argument of Varopoulos [32], we can apply The-orem 3.1 to give a Sobolev inequality proved by Li and Wang (see Corollary 5.13 in [21])on complete gradient shrinking Ricci solitons. Here we only provide the result withoutproof. The detailed proof could follow the argument of Theorem 11.6 in [18] by using theSchr¨odinger operator L instead of the Laplace operator. HARP GAUSSIAN UPPER BOUNDS FOR SCHR ¨ODINGER HEAT KERNEL 11
Proposition 3.5.
Let ( M, g, f ) be an n -dimensional complete gradient shrinking Ricci soli-ton satisfying (1.1) and (1.2) with scalar curvature R bounded above by a constant. Thenthere exists a constant depending only on n such that Z B p ( r ) u nn − dv ! n − n ≤ C ( n ) e − µn Z B p ( r ) (cid:0) |∇ u | + a R u (cid:1) dv for each compactly supported smooth function u with supported in B p ( r ) , where p ∈ M and r > . Here µ := µ ( g, is the entropy functional of Perelman, and a is a constant with a ≥ . Gaussian upper bound
In this section, we will follow the argument of Grigor’yan [14] to prove Theorem 1.1. Let u ( x, t ) be a solution to the Dirichlet boundary condition for the equation ( ∂ t − L ) u = 0in Ω × (0 , T ) with a compact set K in Ω, where Ω ⊂ M . We consider two integrals of asolution u ( x, t ) I ( t ) := Z Ω u ( x, t ) dv and E D ( t ) := Z Ω u ( x, t ) exp (cid:18) d ( x, K ) Dt (cid:19) dv, where D is a positive number. Obviously, I ( t ) ≤ E D ( t ). In the following, we will prove areverse inequality in some ways. Proposition 4.1.
Let u ( x, t ) be a solution to the Dirichlet condition for the equation ( ∂ t − L ) u = 0 . Assume that for any t ∈ (0 , T ) , I ( t ) ≤ e − µ (8 πt ) n . Then, for any γ > , D > and for all t ∈ (0 , T ) , E D ( t ) ≤ e − µ (8 πδt ) n for some δ = δ ( D, γ ) > . Here µ is the Perelman’s entropy functional. In order to prove this proposition, we start from a useful lemma.
Lemma 4.2.
Under the hypotheses of Proposition 4.1, for any γ > , there exists D = D ( γ ) > , such that I R ( t ) ≤ e − µ (8 π tγ ) n exp (cid:18) − R D t (cid:19) for all R > and t ∈ (0 , T ) , where µ is the Perelman’s entropy functional, and I R ( t ) := Z Ω \ B ( K,R ) u ( x, t ) dv. Here B ( K, R ) denotes the open R -neighbourhood of set K . Proof of Lemma 4.2.
To prove the estimate, we first claim that I R ( t ) has a comparisonresult:(4.1) I R ( t ) ≤ I r ( τ ) + e − µ (8 πτ ) n exp (cid:18) − ( R − r ) t − τ ) (cid:19) for R > r and t > τ . This claim follows by the integral monotonicity, which says that: thefollowing function Z Ω u ( x, t ) e ξ ( x,t ) dv, is decreasing in t ∈ (0 , T ). Here function ξ ( x, t ) is defined as ξ ( x, t ) := d ( x )2( t − s )for s > t , where d ( x ) is a distance function defined by d ( x ) = (cid:26) R − d ( x, K ) if x ∈ B ( K, R ) , x / ∈ B ( K, R ) . In fact ddt Z Ω u ( x, t ) e ξ ( x,t ) dv = Z Ω u ξ t e ξ dv + Z Ω uu t e ξ dv ≤ − Z Ω u |∇ ξ | e ξ dv + Z Ω u (∆ u − a R u ) e ξ dv ≤ − Z Ω u |∇ ξ | e ξ dv − Z Ω ∇ u ∇ ( ue ξ ) dv = − Z Ω ( u ∇ ξ + 2 ∇ u ) e ξ dv ≤ . We now continue to prove the claim (4.1). By the integral monotonicity, we have(4.2) Z Ω u ( x, t ) e − d x )2( s − t ) dv ≤ Z Ω u ( x, τ ) e − d x )2( s − τ ) dv for s > t > τ . Notice that, by the definition of d ( x ), on one hand, Z Ω u ( x, t ) e − d x )2( s − t ) dv = Z Ω \ B ( K,R ) u ( x, t ) e − d x )2( s − t ) dv + Z B ( K,R ) u ( x, t ) e − d x )2( s − t ) dv ≥ Z Ω \ B ( K,R ) u ( x, t ) dv = I R ( t ); HARP GAUSSIAN UPPER BOUNDS FOR SCHR ¨ODINGER HEAT KERNEL 13 on the other hand, Z Ω u ( x, τ ) e − d x )2( s − τ ) dv = Z Ω \ B ( K,r ) u ( x, τ ) e − d x )2( s − τ ) dv + Z B ( K,r ) u ( x, τ ) e − d x )2( s − τ ) dv ≤ Z Ω \ B ( K,r ) u ( x, τ ) dv + Z B ( K,r ) u ( x, τ ) e − ( R − r )22( s − τ ) dv = I r ( t ) + exp (cid:18) − ( R − r ) s − τ ) (cid:19) Z B ( K,r ) u ( x, τ ) dv, where R > r . Combining these estimates, (4.2) becomes I R ( t ) ≤ I r ( τ ) + exp (cid:18) − ( R − r ) s − τ ) (cid:19) Z B ( K,r ) u ( x, τ ) dv, and hence claim (4.1) follows by letting s → t + and the assumption of Proposition 4.1.Then, we will apply (4.1) to prove (4.2) by the iteration technique. Choose R k and t k asfollows: R k = (cid:18)
12 + 1 k + 2 (cid:19) R, t k = tγ k , where γ > R k , t k ) and ( R k +1 , t k +1 ) to (4.1) and getthe following iterated inequality I R k ( t k ) ≤ I R k +1 ( t k +1 ) + e − µ (8 πt k +1 ) n exp (cid:18) − ( R k − R k +1 ) t k − t k +1 ) (cid:19) . Sum up the above inequalities over all k = 0 , , ... , I R ( t ) ≤ ∞ X k =0 e − µ (8 πt k +1 ) n exp (cid:20) − ( R k − R k +1 ) t k − t k +1 ) (cid:21) , where we used a fact that lim k →∞ I R k ( t k ) = Z Ω \ B ( K,R/ u ( x, dv = 0by the Dirichlet boundary condition of u . Since t k +1 = tγ k +1 , R k − R k +1 ≥ R ( k + 3) and t k − t k +1 = γ − γ k +1 t, then we have I R ( t ) ≤ e − µ (8 πt ) n ∞ X k =0 exp (cid:20) ( k + 1) n γ − γ k +1 ( γ − k + 3) · R t (cid:21) . Noticing a easy fact that γ k +1 grows in k much faster than denominator ( k + 3) whenever γ >
1, we let m = m ( γ ) := inf k ≥ γ k +1 ( γ − k + 2)( k + 3) . Then, I R ( t ) ≤ e − µ (8 πt ) n ∞ X k =0 exp (cid:20) ( k + 1) n γ − m ( k + 2) · R t (cid:21) = e − µ (8 πt ) n exp (cid:18) − m R t (cid:19) ∞ X k =0 exp (cid:20) ( k + 1) (cid:18) n γ − m R t (cid:19)(cid:21) . We shall further estimate the right hand side of the above inequality. When n γ − m R t ≤ − ln 2 , we have I R ( t ) ≤ e − µ (8 πt ) n exp (cid:18) − m R t (cid:19) ∞ X k =0 − ( k +1) = e − µ (8 πt ) n exp (cid:18) − m R t (cid:19) . When n γ − m R t > − ln 2 , we use the definitions of I R ( t ) and I ( t ) and have that I R ( t ) ≤ I ( t ) ≤ e − µ (8 πt ) n ≤ e − µ (8 πt ) n exp (cid:18) n γ + ln 2 − m R t (cid:19) = 2 e − µ (8 π tγ ) n exp (cid:18) − m R t (cid:19) . Therefore, in any case I R ( t ) ≤ e − µ (8 π tγ ) n exp (cid:18) − m R t (cid:19) , where m = m ( γ ) < (cid:3) Now we apply Lemma 4.2 to give the proof of Proposition 4.1.
Proof of Proposition 4.1. Step One : we show that for D ≥ D and for all t > E D ( t ) ≤ e − µ (8 π tγ ) n . HARP GAUSSIAN UPPER BOUNDS FOR SCHR ¨ODINGER HEAT KERNEL 15
By the definition of E D ( t ), we split E D ( t ) into two terms: E D ( t ) = Z Ω u exp (cid:18) d ( x, K ) Dt (cid:19) dv = Z { d ( x,K ) ≤ R } u exp (cid:18) d ( x, K ) Dt (cid:19) dv + ∞ X k =0 Z { k R ≤ d ( x,K ) ≤ k +1 R } u exp (cid:18) d ( x, K ) Dt (cid:19) dv ≤ Z Ω u exp (cid:18) R Dt (cid:19) dv + ∞ X k =0 Z { k R ≤ d ( x,K ) ≤ k +1 R } u exp (cid:18) d ( x, K ) Dt (cid:19) dv ≤ e − µ (8 πt ) n exp (cid:18) R Dt (cid:19) + ∞ X k =0 Z { k R ≤ d ( x,K ) ≤ k +1 R } u exp (cid:18) d ( x, K ) Dt (cid:19) dv. By Lemma 4.2, the k -th factor in the above sum term can be estimated by Z { k R ≤ d ( x,K ) ≤ k +1 R } u exp (cid:18) d ( x, K ) Dt (cid:19) dv ≤ exp (cid:18) k +1 R Dt (cid:19) Z Ω \ B ( K, k R ) u dv ≤ e − µ (8 π tγ ) n exp (cid:18) k +1 R Dt − k R D t (cid:19) ≤ e − µ (8 π tγ ) n exp (cid:18) − k R Dt (cid:19) where we used D ≥ D . Therefore, E D ( t ) ≤ e − µ (8 πt ) n exp (cid:18) R Dt (cid:19) + 2 e − µ (8 π tγ ) n ∞ X k =0 exp (cid:18) − k R Dt (cid:19) for any R >
0. In particular, we choose R = Dt ln 2 and get E D ( t ) ≤ e − µ (8 πt ) n + 2 e − µ (8 π tγ ) n ∞ X k =0 − k ≤ e − µ (8 π tγ ) n . Step Two : In the rest, it suffices to prove the case 2 < D < D . In the proof of Lemma4.2, we prove R Ω u ( x, t ) e d x )2( t − s ) dv , s > t , is decreasing in t , which implies that for any s > R Ω u ( x, t ) e d x,K )2( t + s ) dv is decreasing in t . Therefore, for any τ ∈ (0 , t ), we have Z Ω u ( x, t ) e d x,K )2( t + s ) dv ≤ Z Ω u ( x, τ ) e d x,K )2( τ + s ) dv. Given 2 < D < D and t , we let s = D − t and τ = D − D − t < t in the above inequality, E D ( t ) ≤ E D ( τ )for any τ ∈ (0 , t ). By step one, we have proven E D ( τ ) ≤ e − µ (8 π τγ ) n . Hence E D ( t ) ≤ e − µ (cid:16) π ( D − D − · tγ (cid:17) n . This implies Proposition 4.1 by letting δ = δ ( D, γ ) = D − D − γ − . (cid:3) Next, we will apply Proposition 4.1 to prove Theorem 1.1.
Proof of Theorem 1.1.
By the semigroup property of the Schr¨odinger heat kernel, we have H R ( x, y, t ) = Z M H R ( x, z, t/ H R ( z, y, t/ dv ( z ) . Then by the triangle inequality d ( x, y ) ≤ d ( x, z ) + d ( y, z )), for any a positive constant D , we furthermore have H R ( x, y, t ) ≤ Z M H R ( x, z, t/ e d x,z ) Dt H R ( z, y, t/ e d y,z ) Dt e − d x,y )2 Dt dv ( z ) ≤ e − d x,y )2 Dt "Z M (cid:18) H R ( x, z, t/ e d x,z ) Dt (cid:19) dv ( z ) / × "Z M (cid:18) H R ( y, z, t/ e d y,z ) Dt (cid:19) dv ( z ) / . If we set E D ( x, t ) := Z M ( H R ( x, z, t )) e d x,z ) Dt dv ( z ) , then we have a simple expression(4.3) H R ( x, y, t ) ≤ p E D ( x, t/ E D ( y, t/
2) exp (cid:18) − d ( x, y )2 Dt (cid:19) , which always holds on ( M, g, f ).We take an increasing sequence of pre-compact regions Ω k ⊂ M , k = 1 , , ... , exhausting M , and in each Ω k we construct the Dirichlet heat kernel H RΩ k ( x, y, t ) for the equation( ∂ t − L ) u = 0. Then by the maximum principle, we have0 ≤ H RΩ k ≤ H RΩ k +1 ≤ H R . Now we will apply Proposition 4.1 to estimate E D, Ω k ( x, t ), where E D, Ω k ( x, t ) := Z Ω k ( H R ( x, z, t )) e d x,z ) Dt dv ( z ) . HARP GAUSSIAN UPPER BOUNDS FOR SCHR ¨ODINGER HEAT KERNEL 17
Let u ( z, t ) = H RΩ k ( x, z, t ) be a solution to the Dirichlet boundary condition for the equation( ∂ t − L ) u = 0 in Ω k × (0 , T ) with K = { x } in Ω. We observe that I ( t ) = Z Ω k u ( z, t ) dv ( z )= Z Ω k H RΩ k ( x, z, t ) H RΩ k ( z, x, t ) dv ( z )= H RΩ k ( x, x, t ) ≤ e − µ (8 πt ) n , where we used Theorem 3.1 in the last inequality. So we can apply Proposition 4.1 to getthat for any γ > D > t ∈ (0 , T ), we have E D, Ω k ( x, t ) = Z Ω k u ( z, t ) e d x,z ) Dt dv ( z ) ≤ e − µ (8 πδt ) n for some δ = δ ( D, γ ) >
0. Letting k → ∞ , we then have E D ( x, t/ ≤ e − µ (4 πδt ) n . By a similar argument to the point y ∈ M , we have E D ( y, t/ ≤ e − µ (4 πδt ) n . Substituting the above two estimates into (4.3) completes the proof of Theorem 1.1. (cid:3) Green’s function estimate
In this section, we will apply Schr¨odinger heat kernel estimates to give the Green’s functionestimate of the Schr¨odinger operator on complete gradient shrinking Ricci solitons; seeTheorem 1.4. Recall that Malgrange [22] proved that any Riemannian manifold admits aGreen’s function G ( x, y ) := Z ∞ H ( x, y, t ) dt if the integral on the right hand side converges, where H ( x, y, t ) denotes the heat kernelof the Laplace operator. Varopoulos [31] showed that any complete manifold ( M, g ) has apositive Green’s function only if(5.1) Z ∞ tV p ( t ) dt < ∞ , where V p ( t ) denotes the volume of the geodesic ball of radius t with center at p ∈ M . Whenthe Ricci curvature of manifolds is nonnegative, Varopoulos [31] and Li-Yau [20] proved(5.1) is a sufficient and necessary condition for the existence of positive Green’s function. On an n -dimensional complete gradient shrinking Ricci soliton ( M, g, f ), letting H R ( x, y, t )be a Schr¨odinger heat kernel of the operator L = − ∆ + a R with a ≥ , the Green’s functionof L is defined by G R ( x, y ) = Z ∞ H R ( x, y, t ) dt if the integral on the right hand side converges. By the Schr¨odinger heat kernel H R ( x, y, t )estimates, it is easy to get an upper estimate for the Green’s function of L , which is similarto Li-Yau estimate [20] of the classical Green’s function. Proof of Theorem 1.4.
Using the definition of G R ( x, y ) and Theorem 1.1, we have G R ( x, y ) = Z ∞ H R ( x, y, t ) dt = Z r H R ( x, y, t ) dt + Z ∞ r H R ( x, y, t ) dt ≤ Z r H R ( x, y, t ) dt + Ae − µ (4 π ) n/ Z ∞ r t − n dt ≤ Ae − µ (4 π ) n/ "Z r t − n exp (cid:18) − r ct (cid:19) dt + Z ∞ r t − n dt . Notice that for the first term of the right hand side of the above inequality, letting s = r /t ,where r < s < ∞ , we get Z r t − n exp (cid:18) − r ct (cid:19) dt = Z ∞ r (cid:18) r s (cid:19) − n exp (cid:18) − scr (cid:19) r s ds = Z ∞ r s − n (cid:16) sr (cid:17) n − exp (cid:18) − scr (cid:19) ds ≤ c ( n ) Z ∞ r s − n ds, where in the last line we have used the fact that the function x n − e − x/c ( c >
4) is boundedfrom above. Therefore, for n ≥ G R ( x, y ) ≤ C ( n ) Ae − µ (4 π ) n/ Z ∞ r t − n dt = 2 C ( n ) Ae − µ ( n − π ) n/ r n − . and the result follows. (cid:3) eigenvalue estimate In this section we apply Gaussian upper bounds of the Schr¨odinger heat kernel H R ( x, y, t )on compact gradient shrinking Ricci solitons to get the eigenvalue estimates of the Schr¨odingeroperator L ; see Theorem 1.6. The proof is essential parallel to the Li-Yau’s Laplace situationon manifolds [20]. HARP GAUSSIAN UPPER BOUNDS FOR SCHR ¨ODINGER HEAT KERNEL 19
Proof of Theorem 1.6.
By Theorem 3.1, the Schr¨odinger heat kernel of the operator L hasan upper bound(6.1) H R ( x, y, t ) ≤ e − µ (4 πt ) n . Notice that the heat Schr¨odinger kernel can be written as H R ( x, y, t ) = ∞ X i =1 e − λ i t ϕ i ( x ) ϕ i ( y ) , where ϕ i is the eigenfunction to the corresponding eigenvalue λ i with k ϕ i k L = 1. Integratingboth sides of (6.1), we have ∞ X i =1 e − λ i t ≤ e − µ (4 πt ) n V ( M ) , where V ( M ) is the volume of the manifold M . Hence, ke − λ k t ≤ e − µ (4 πt ) n V ( M ) , which further implies(6.2) ke µ V ( M ) ≤ e λ k t (4 πt ) − n for any t >
0. It is easy to see that function e λ k t (4 πt ) − n takes its minimum at t = n λ k . Plugging this point to (6.2) gives the lower bound of λ k . (cid:3) References
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