Universal inequalities for eigenvalues. of elliptic operators in divergence. form on domains in complete. noncompact Riemannian manifolds
|
|
- Gavin Godfrey Jackson
- 6 years ago
- Views:
Transcription
1 Theoretical athematics & Applications, vol.3, no., 03, ISSN: print, online Scienpress Ltd, 03 Universal inequalities for eigenvalues of elliptic operators in divergence form on domains in complete noncompact Riemannian manifolds Yanli Li and Feng Du Abstract In this paper, we study the eigenvalue problem of elliptic operators in divergence form, and obtain some universal inequalities for eigenvalues of elliptic operators in divergence form on domains in complete simple connected noncompact Riemannian manifolds admitting special functions which include hyperbolic space. Especially, by using our universal inequalities, we can get the different universal inequalities including Yang inequality. athematics Subject Classification: 35P5, 53C4, 58G5 Keywords: Elliptic operators in divergence form, eigenvalues, universal inequalities, complete noncompact Riemannian manifolds, hyperbolic space School of Electronic and Information Science, Jingchu University of Technology, Hubei Jingmen , P.R. China. School of athematics and Physics Science, Jingchu University of Technology, Hubei Jingmen , P.R. China. Article Info: Received : April 0, 03. Revised : ay 30, 03 Published online : June 5, 03
2 40 Universal inequalities for eigenvalues Introduction Let be a bounded domain in an n-dimensional complete Riemannian manifold. Let be the Laplacian operator acting on functions on and consider the following eigenvalues problem for the Laplacian operator u = λu, in, u = 0, on, it is known that this eigenvalue problem has a discrete spectrum, 0 < λ < λ λ k,. where each eigenvalue is repeated with its multiplicity. When = R n, = n, Payne-Pólya-Weinberger [] in 956 proved x i λ k+ λ k 4 kn In 980, Hile-Protter [9] strengthened., and proved λ i.. kn 4 λ i..3 λ k+ λ i In 99, Yang [3] gave the following much stronger inequality λ k+ λ i 4 n λ k+ λ i λ i..4 From inequality.3, we can get a weaker but explicit form λ k+ + 4 λ i..5 n k These inequalities are called universal inequalities because they do not involve domain dependence. In the following, we introduce an elliptic operator in divergence form in Riemannian manifold. Let,, be an n-dimensional compact Riemannian manifold with boundary possibly empty. Let A : EndT be a
3 Yanli Li and Feng Du 4 smooth symmetric and positive definite section of the bundle of all endomorphisms of T. Let V be a nonnegative continuous function on. Denote by and the Laplacian and the gradient operator of respectively. Define Lu = diva u + V u,.6 where for a vector field X on, divx denotes the divergence of X. The operator L defined in.5 is an elliptic operator in divergence form. It is easy to see that the Laplacian operator and Schrödinger operator are it s special cases. In 00, do Carmo-Wang-Xia [6] considered the eigenvalue problem of the elliptic operator in divergence form with weight such that Lu = λρu, in, u = 0, on,.7 where ρ is a weight function which is positive and continuous on. They got a Yang type inequality λ k+ λ i 4ξ ρ nρ λ k+ λ i ξ λ i V 0 + n H0,.8 ρ 4ρ where ξ I A and tra nξ throughout, ρ ρx ρ, x, I is the identity map, ξ, ξ, ρ, ρ are positive constants, H 0 = max H, V 0 = min V, and H is the mean curvature of immersed into some Euclidean space R N. Recently, Du-Wu-Li[7] considered the problem.7 on complete Riemannian manifolds, they proved the inequality fλ i ρ nρ nξ ξ gλ i fλ i λ k+ λ i gλ i n where f, g I λk+,s = sup ξ 4 H V vector field. λ i + S,.9 ρ, and H is the mean curvature In this paper, we will consider the eigenvalue problem.7 on bounded domains in the complete simply connected noncompact Riemannian manifolds. Then we obtain.
4 4 Universal inequalities for eigenvalues Theorem.. Let,, be an nn 3-dimensional complete noncompact simply connected Riemannian manifold with Sectional curvature Sec satisfying K Sec k. For a bounded domain, Let A : EndT be a smooth symmetric and positive definite section of the bundle of all endomorphisms of T, and assume that ξ I A ξ I throughout, x, ρ ρx ρ, here ξ, ξ, ρ, ρ are positive constants. Let λ i be the i th eigenvalue of the eigenvalue problem.7, then for any f, g I λk+, we have fλ i { } { gλ i fλ i λ k+ λ i gλ i } ρ ξ ρ λ i V 0 n k + n K ξ.0 where V 0 = min x V x. From Theorem., we can get the following. Corollary.. Under the assumption of Theorem., if is a hyperbolic space H n, we have fλ i { ρ ξ { where V 0 = min x V x. gλ i } fλ i λ k+ λ i gλ i } ρ λ i V 0 n., ξ Remark. Taking fλ i, gλ i = λ k+ λ i, λ k+ λ i in., we can get a Yang inequality that λ k+ λ i ρ ξ λ k+ λ i ρ λ i V 0 n ξ So, by choosing different fλ i, gλ i, we can get different universal inequalities.
5 Yanli Li and Feng Du 43 Preliminaries [8]. In this section, firstly, we shall introduce a family of couples of functions Definition.. Let λ > 0, a couple f, g of functions defined on [0, λ] belongs to I λ as that i f and g are positive, ii f and g satisfy the following condition, for any x, y [0, λ], such that x y, fx fy fx + x y gxλ x + fy gx gy 0. gyλ y x y. We can easily find that gx is a nonincreasing function. Li[7]. In the following, we will introduce a lemma which is obtained by Du-Wu- Lemma.. Let,, be an n-dimensional compact Riemannian manifold with boundary possibly empty. Let λ i be the i th eigenvalue of the eigenvalue problem of elliptic operators in divergence form with weight ρ such that Lu = λρu in, u = 0 on, and u i be the orthonormal eigenfunction corresponding to λ i, that is, Lu i = λ i ρu i in, and u i = 0 on, ρu i u j = δ ij, i, j =,, Then for any h C and f, g I λk+, we have fλ i u i h δ gλ i u i h, A h + u i, h + ρ u i h fλ i δλ k+ λ i gλ i,.
6 44 Universal inequalities for eigenvalues where δ is any positive constant and f = f. From this Lemma, if we can find a nice function on bounded domains in complete Riemannian manifolds and take it in. and.3, we can get universal inequalities in complete Riemannian manifolds. In the following, we will introduce a function on a bounded domain in complete noncompact simple connected Riemannian manifoldsfor the more details about this function, we refer [5]. Let,, be an n-dimensional complete noncompact Riemannian manifold, and be a bounded connected domain in. Sec is the section curvature satisfying K Sec k, where k, K is constants and 0 k K. When p is not in, define the distance function rx = dx, p, then n k coshkr sinhkr because of r r = Hess r Ric r, r, so r n K coshkr sinhkr,.3 r r n K coshkr sinhkr n k,.4 where Hess, Ric are the Hessian operator and Ricci curvature operator, respectively. 3 Proofs of the main results In this section, we will give the proofs of Theorem. and Corollary.. Proof of Theorem.. Taking h = r in the., we can get fλ i u i δ gλ i u i r, A r + u i, r + ρ u i r fλ i δλ k+ λ i gλ i, 3.
7 Yanli Li and Feng Du 45 Because of ρ ρx ρ, ξ I A ξ I, and., we have u i 3. ρ ρ ξ r, A r ξ. 3.3 Take 3. and 3.3 into 3., we have fλ i ξ fλ i δ gλ i + ρ ρ 4δλ k+ λ i gλ i u i, r + u i r ρ 3.4 It is known that λ i = which yields u i Lu i = = u i diva u i + V u i Aui, u i + V u i ξ u i + V u i, u i λ i V ξ ρ From.4 and.5, we can get u i, r + u i r ρ = ρ r, u i + u i r 4 r, u i u i r u i r, r ρ u i u i r u i r r ρ λ i V 0 n k u cosh kr i ρ ξ ρ ρ sinh kr n K + u cosh Kr n k i ρ sinh Kr ρ ρ λ i V 0 n k n K + ρ ξ ρ ρ ρ ρ ρ n ρ u i k n sinh + kr ρ u i K sinh Kr 3.6
8 46 Universal inequalities for eigenvalues Since K k 0, r > 0, we obtain Since n 3, we get K sinhkr k sinhkr, 3.7 n k sinh kr n K so, by , we can get ρ u i r + r, u i λ i V 0 ρ ξ ρ Taking 3.0 into 3.4, we obtain sinh Kr n n 3 k n k ρ ρ + sinh kr 0, 3.8 n K ρ ρ 3.9 ρ fλ i ξ ρ δ gλ i + ρ ξ λ i V 0 ρ fλ i 4δλ k+ λ i gλ i n k + ρ ρ n K ρ ρ 3.0, above inequality implies that fλ i ρ ξ δ ρ gλ i + fλ i 4δλ k+ λ i gλ i ρ λ i V 0 n k + n K ρ ξ 3. δ = 4ρ ξ k 3. becomes fλ i λ k+ λ i gλ i k gλ i ξ ρ λ i V 0 n k + n K fλ i { } { gλ i fλ i λ k+ λ i gλ i } ρ ξ ρ λ i V 0 n k + n K ξ 3. This completes the proof of Theorem..
9 Yanli Li and Feng Du 47 Proof of Corollary.. Taking K = k = in 3., we immediately obtain.. ACKNOWLEDGEENTS. The research work is supported by Key Laboratory of Applied athematics of Hubei Province and The research project of Jingchu University of Technology. References [].S. Ashbaugh, Isoperimetric and unversal inequalities for eigenvalues, In: Davies, E.B., Safalov, Y.eds. Spectral theory and geometry Edinburgh, 998, London ath. Soc. Lecture Notes, Cambridge University Press, Cambridge 999, pp [] D. Chen and Q.. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. ath. Soc. Jpn., 60, 008, [3] Q.. Cheng, T. Ichikawa and S. ametsuka, Estimates for eigenvalues of the poly-laplacian with any order, Communication in Contemporary athematics, 4, 009, [4] Q.. Cheng and H.C. Yang, Estimates on eigenvalues of Laplacian, ath. Ann., 33, 005, [5] D. Chen, T. Zheng and. Lu, Eigenvalue estimates on domain in complete noncompact Riemannian manifolds, Pacific Journal of athematics, 55, 0, [6].P. do Carmo, Q. Wang and C. Xia, Inequalities for eigenvalues of elliptic operators in divergence form on Riemannian manifolds. Annali di athematica, 89, 00, [7] F. Du, C. Wu and G. Li, Extrinsic estimates for eigenvalues of elliptic operators in divergence form, Submitted to Umzh. mathe.
10 48 Universal inequalities for eigenvalues [8] S. Ilias and O. akhoul, Universal inequalities for the eigenvalues of a power of the Laplace operator, anuscripta ath., 3, 00, [9] G.N. Hile and.h. Protter, Inequalities for eigenvalues of the Laplacian, Indiana Univ. ath. J., 9, 980, [0] E.. Harrel and J. Stubbe, On trace inequalities and the universal eigenvalue estimates for some partial differntial operators, Trans. Am. ath. Soc., 349, 997, [] L.E. Payne, G. Pólya and H.F. Weinberger, On the ratio of consecutive eigenvalues, J. ath. Phys., 35, 956, [] Q. Wang and C. Xia, Universal bounds for eigenvalues of Schrödinger operator on Riemannian manifolds, Ann. Acad. Sci. Fenn. ath., 33, 008, [3] H.C. Yang, An estimate of the difference between consecutive eigenvalues, preprint IC/9/60 of ICTP, Trieste, 99.
On extensions of Myers theorem
On extensions of yers theorem Xue-ei Li Abstract Let be a compact Riemannian manifold and h a smooth function on. Let ρ h x = inf v =1 Ric x v, v 2Hessh x v, v. Here Ric x denotes the Ricci curvature at
More informationarxiv: v1 [math.dg] 25 Nov 2009
arxiv:09.4830v [math.dg] 25 Nov 2009 EXTENSION OF REILLY FORULA WITH APPLICATIONS TO EIGENVALUE ESTIATES FOR DRIFTING LAPLACINS LI A, SHENG-HUA DU Abstract. In this paper, we extend the Reilly formula
More informationarxiv:math/ v1 [math.dg] 7 Jun 2004
arxiv:math/46v [math.dg] 7 Jun 4 The First Dirichlet Eigenvalue and Li s Conjecture Jun LING Abstract We give a new estimate on the lower bound for the first Dirichlet eigenvalue for the compact manifolds
More informationUniversal patterns in spectra of differential operators. Copyright 2008 by Evans M. Harrell II.
Universal patterns in spectra of differential operators Copyright 2008 by Evans M. Harrell II. This lecture will not involve snakes, machetes, or other dangerous objects. Laplace, Laplace-Beltrami, and
More informationA GENERALIZATION OF A LEVITIN AND PARNOVSKI UNIVERSAL INEQUALITY FOR EIGENVALUES
A GENERALIZATION OF A LEVITIN AND PARNOVSKI UNIVERSAL INEQUALITY FOR EIGENVALUES Said Ilias, akhoul Ola To cite this version: Said Ilias, akhoul Ola. A GENERALIZATION OF A LEVITIN AND PARNOVSKI UNI- VERSAL
More informationSum rules and semiclassical limits for quantum Hamiltonians on surfaces, periodic structures, and graphs
Sum rules and semiclassical limits for quantum Hamiltonians on surfaces, periodic structures, and graphs Evans Harrell Georgia Tech www.math.gatech.edu/~harrell Centre Bernoulli École Polytechnique Fédérale
More informationLiouville Properties for Nonsymmetric Diffusion Operators. Nelson Castañeda. Central Connecticut State University
Liouville Properties for Nonsymmetric Diffusion Operators Nelson Castañeda Central Connecticut State University VII Americas School in Differential Equations and Nonlinear Analysis We consider nonsymmetric
More informationThe eigenvalue problem in Finsler geometry
The eigenvalue problem in Finsler geometry Qiaoling Xia Abstract. One of the fundamental problems is to study the eigenvalue problem for the differential operator in geometric analysis. In this article,
More informationUNIVERSAL INEQUALITIES FOR THE EIGENVALUES OF LAPLACE AND SCHRÖDINGER OPERATORS ON SUBMANIFOLDS
UNIVERSAL INEQUALITIES FOR THE EIGENVALUES OF LAPLACE AND SCHRÖDINGER OPERATORS ON SUBANIFOLDS AHAD EL SOUFI, EVANS. HARRELL II, AND SAÏD ILIAS Abstract. We establish inequalities for the eigenvalues of
More informationSum rules and semiclassical limits for the spectra of some elliptic PDEs and pseudodifferential operators
Sum rules and semiclassical limits for the spectra of some elliptic PDEs and pseudodifferential operators Evans Harrell Georgia Tech www.math.gatech.edu/~harrell Institut É. Cartan Univ. H. Poincaré Nancy
More informationGradient Estimates and Sobolev Inequality
Gradient Estimates and Sobolev Inequality Jiaping Wang University of Minnesota ( Joint work with Linfeng Zhou) Conference on Geometric Analysis in honor of Peter Li University of California, Irvine January
More informationFOUR-DIMENSIONAL GRADIENT SHRINKING SOLITONS WITH POSITIVE ISOTROPIC CURVATURE
FOUR-DIENIONAL GRADIENT HRINKING OLITON WITH POITIVE IOTROPIC CURVATURE XIAOLONG LI, LEI NI, AND KUI WANG Abstract. We show that a four-dimensional complete gradient shrinking Ricci soliton with positive
More informationSum rules and semiclassical limits for quantum Hamiltonians on surfaces, periodic structures, and graphs.
Sum rules and semiclassical limits for quantum Hamiltonians on surfaces, periodic structures, and graphs. Evans Harrell Georgia Tech www.math.gatech.edu/~harrell Mathematical aspects of quantum transport
More informationA Note on Cohomology of a Riemannian Manifold
Int. J. Contemp. ath. Sciences, Vol. 9, 2014, no. 2, 51-56 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2014.311131 A Note on Cohomology of a Riemannian anifold Tahsin Ghazal King Saud
More informationJeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi
Bull. Korean Math. Soc. 40 (003), No. 3, pp. 411 43 B.-Y. CHEN INEQUALITIES FOR SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi Abstract. Some B.-Y.
More informationCalderón-Zygmund inequality on noncompact Riem. manifolds
The Calderón-Zygmund inequality on noncompact Riemannian manifolds Institut für Mathematik Humboldt-Universität zu Berlin Geometric Structures and Spectral Invariants Berlin, May 16, 2014 This talk is
More informationEssential Spectra of complete manifolds
Essential Spectra of complete manifolds Zhiqin Lu Analysis, Complex Geometry, and Mathematical Physics: A Conference in Honor of Duong H. Phong May 7, 2013 Zhiqin Lu, Dept. Math, UCI Essential Spectra
More informationThe volume growth of complete gradient shrinking Ricci solitons
arxiv:0904.0798v [math.dg] Apr 009 The volume growth of complete gradient shrinking Ricci solitons Ovidiu Munteanu Abstract We prove that any gradient shrinking Ricci soliton has at most Euclidean volume
More informationSpacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds
Spacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds Alfonso Romero Departamento de Geometría y Topología Universidad de Granada 18071-Granada Web: http://www.ugr.es/
More information1-TYPE AND BIHARMONIC FRENET CURVES IN LORENTZIAN 3-SPACE *
Iranian Journal of Science & Technology, Transaction A, ol., No. A Printed in the Islamic Republic of Iran, 009 Shiraz University -TYPE AND BIHARMONIC FRENET CURES IN LORENTZIAN -SPACE * H. KOCAYIGIT **
More informationand finally, any second order divergence form elliptic operator
Supporting Information: Mathematical proofs Preliminaries Let be an arbitrary bounded open set in R n and let L be any elliptic differential operator associated to a symmetric positive bilinear form B
More informationGeometric bounds for Steklov eigenvalues
Geometric bounds for Steklov eigenvalues Luigi Provenzano École Polytechnique Fédérale de Lausanne, Switzerland Joint work with Joachim Stubbe June 20, 2017 luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues
More informationSELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
More informationQing-Ming Cheng and Young Jin Suh
J. Korean Math. Soc. 43 (2006), No. 1, pp. 147 157 MAXIMAL SPACE-LIKE HYPERSURFACES IN H 4 1 ( 1) WITH ZERO GAUSS-KRONECKER CURVATURE Qing-Ming Cheng and Young Jin Suh Abstract. In this paper, we study
More informationComplete spacelike hypersurfaces with positive r-th mean curvature in a semi-riemannian warped product
DOI: 10.1515/auom-2015-0041 An. Şt. Univ. Ovidius Constanţa Vol. 23(2),2015, 259 277 Complete spacelike hypersurfaces with positive r-th mean curvature in a semi-riemannian warped product Yaning Wang and
More information338 Jin Suk Pak and Yang Jae Shin 2. Preliminaries Let M be a( + )-dimensional almost contact metric manifold with an almost contact metric structure
Comm. Korean Math. Soc. 3(998), No. 2, pp. 337-343 A NOTE ON CONTACT CONFORMAL CURVATURE TENSOR Jin Suk Pak* and Yang Jae Shin Abstract. In this paper we show that every contact metric manifold with vanishing
More informationUNIQUENESS RESULTS ON SURFACES WITH BOUNDARY
UNIQUENESS RESULTS ON SURFACES WITH BOUNDARY XIAODONG WANG. Introduction The following theorem is proved by Bidaut-Veron and Veron [BVV]. Theorem. Let (M n, g) be a compact Riemannian manifold and u C
More informationarxiv:math/ v1 [math.dg] 19 Nov 2004
arxiv:math/04426v [math.dg] 9 Nov 2004 REMARKS ON GRADIENT RICCI SOLITONS LI MA Abstract. In this paper, we study the gradient Ricci soliton equation on a complete Riemannian manifold. We show that under
More informationWeighted Graph Laplacians and Isoperimetric Inequalities
Weighted Graph Laplacians and Isoperimetric Inequalities Fan Chung University of California, San Diego La Jolla, California 92093-0112 Kevin Oden Harvard University Cambridge, Massachusetts 02138 Abstract
More informationCOMPLETE SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE
Chao, X. Osaka J. Math. 50 (203), 75 723 COMPLETE SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE XIAOLI CHAO (Received August 8, 20, revised December 7, 20) Abstract In this paper, by modifying Cheng Yau
More informationarxiv: v1 [math.dg] 28 Aug 2017
SOE CLASSIFICATIONS OF BIHARONIC HYPERSURFACES WITH CONSTANT SCALAR CURVATURE SHUN AETA AND YE-LIN OU arxiv:708.08540v [math.dg] 28 Aug 207 Abstract We give some classifications of biharmonic hypersurfaces
More informationRICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS
J. Austral. Math. Soc. 72 (2002), 27 256 RICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS ION MIHAI (Received 5 June 2000; revised 19 February 2001) Communicated by K. Wysocki Abstract Recently,
More informationSharp gradient estimate and spectral rigidity for p-laplacian
Shar gradient estimate and sectral rigidity for -Lalacian Chiung-Jue Anna Sung and Jiaing Wang To aear in ath. Research Letters. Abstract We derive a shar gradient estimate for ositive eigenfunctions of
More informationHYPERSURFACES OF EUCLIDEAN SPACE AS GRADIENT RICCI SOLITONS *
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LXI, 2015, f.2 HYPERSURFACES OF EUCLIDEAN SPACE AS GRADIENT RICCI SOLITONS * BY HANA AL-SODAIS, HAILA ALODAN and SHARIEF
More informationarxiv: v1 [math.dg] 1 Jul 2014
Constrained matrix Li-Yau-Hamilton estimates on Kähler manifolds arxiv:1407.0099v1 [math.dg] 1 Jul 014 Xin-An Ren Sha Yao Li-Ju Shen Guang-Ying Zhang Department of Mathematics, China University of Mining
More informationL -uniqueness of Schrödinger operators on a Riemannian manifold
L -uniqueness of Schrödinger operators on a Riemannian manifold Ludovic Dan Lemle Abstract. The main purpose of this paper is to study L -uniqueness of Schrödinger operators and generalized Schrödinger
More informationSection 6. Laplacian, volume and Hessian comparison theorems
Section 6. Laplacian, volume and Hessian comparison theorems Weimin Sheng December 27, 2009 Two fundamental results in Riemannian geometry are the Laplacian and Hessian comparison theorems for the distance
More informationCONVEXITY OF THE ENTROPY OF POSITIVE SOLUTIONS TO THE HEAT EQUATION ON QUATERNIONIC CONTACT AND CR MANIFOLDS
CONVEXITY OF THE ENTROPY OF POSITIVE SOLUTIONS TO THE HEAT EQUATION ON QUATERNIONIC CONTACT AND CR ANIFOLDS D. VASSILEV Abstract. A proof of the monotonicity of an entropy like energy for the heat equation
More informationThe p-laplacian and geometric structure of the Riemannian manifold
Workshop on Partial Differential Equation and its Applications Institute for Mathematical Sciences - National University of Singapore The p-laplacian and geometric structure of the Riemannian manifold
More informationIsoperimetric Inequalities for the Cauchy-Dirichlet Heat Operator
Filomat 32:3 (218), 885 892 https://doi.org/1.2298/fil183885k Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Isoperimetric Inequalities
More informationNegative sectional curvature and the product complex structure. Harish Sheshadri. Department of Mathematics Indian Institute of Science Bangalore
Negative sectional curvature and the product complex structure Harish Sheshadri Department of Mathematics Indian Institute of Science Bangalore Technical Report No. 2006/4 March 24, 2006 M ath. Res. Lett.
More informationHarmonic Functions on Complete Riemannian Manifolds
* Vol. I?? Harmonic Functions on Complete Riemannian Manifolds Peter Li Abstract We present a brief description of certain aspects of the theory of harmonic functions on a complete Riemannian manifold.
More informationUPPER BOUNDS FOR EIGENVALUES OF THE DISCRETE AND CONTINUOUS LAPLACE OPERATORS
UPPER BOUNDS FOR EIGENVALUES OF THE DISCRETE AND CONTINUOUS LAPLACE OPERATORS F. R. K. Chung University of Pennsylvania, Philadelphia, Pennsylvania 904 A. Grigor yan Imperial College, London, SW7 B UK
More informationM. Ledoux Université de Toulouse, France
ON MANIFOLDS WITH NON-NEGATIVE RICCI CURVATURE AND SOBOLEV INEQUALITIES M. Ledoux Université de Toulouse, France Abstract. Let M be a complete n-dimensional Riemanian manifold with non-negative Ricci curvature
More informationGradient estimates for eigenfunctions on compact Riemannian manifolds with boundary
Gradient estimates for eigenfunctions on compact Riemannian manifolds with boundary Xiangjin Xu Department of athematics Johns Hopkins University Baltimore, D 21218 Abstract The purpose of this paper is
More informationHARNACK INEQUALITY FOR NONDIVERGENT ELLIPTIC OPERATORS ON RIEMANNIAN MANIFOLDS. Seick Kim
HARNACK INEQUALITY FOR NONDIVERGENT ELLIPTIC OPERATORS ON RIEMANNIAN MANIFOLDS Seick Kim We consider second-order linear elliptic operators of nondivergence type which are intrinsically defined on Riemannian
More informationH-convex Riemannian submanifolds
H-convex Riemannian submanifolds Constantin Udrişte and Teodor Oprea Abstract. Having in mind the well known model of Euclidean convex hypersurfaces [4], [5] and the ideas in [1], many authors defined
More informationLogarithmic Harnack inequalities
Logarithmic Harnack inequalities F. R. K. Chung University of Pennsylvania Philadelphia, Pennsylvania 19104 S.-T. Yau Harvard University Cambridge, assachusetts 02138 1 Introduction We consider the relationship
More informationFIRST EIGENVALUES OF GEOMETRIC OPERATOR UNDER THE RICCI-BOURGUIGNON FLOW
J. Inones. ath. Soc. Vol. 24, No. 1 (2018), pp. 51 60. FIRST EIGENVALUES OF GEOETRIC OPERATOR UNDER THE RICCI-BOURGUIGNON FLOW Shahrou Azami Department of athematics, Faculty of Sciences Imam Khomeini
More informationPreprint Preprint Preprint Preprint
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
More informationDeterminant of the Schrödinger Operator on a Metric Graph
Contemporary Mathematics Volume 00, XXXX Determinant of the Schrödinger Operator on a Metric Graph Leonid Friedlander Abstract. In the paper, we derive a formula for computing the determinant of a Schrödinger
More informationDraft version September 15, 2015
Novi Sad J. Math. Vol. XX, No. Y, 0ZZ,??-?? ON NEARLY QUASI-EINSTEIN WARPED PRODUCTS 1 Buddhadev Pal and Arindam Bhattacharyya 3 Abstract. We study nearly quasi-einstein warped product manifolds for arbitrary
More informationCOMPLETE GRADIENT SHRINKING RICCI SOLITONS WITH PINCHED CURVATURE
COMPLETE GRADIENT SHRINKING RICCI SOLITONS WITH PINCHED CURVATURE GIOVANNI CATINO Abstract. We prove that any n dimensional complete gradient shrinking Ricci soliton with pinched Weyl curvature is a finite
More informationA Remark on -harmonic Functions on Riemannian Manifolds
Electronic Journal of ifferential Equations Vol. 1995(1995), No. 07, pp. 1-10. Published June 15, 1995. ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110
More informationRIEMANNIAN SUBMERSIONS NEED NOT PRESERVE POSITIVE RICCI CURVATURE
RIEMANNIAN SUBMERSIONS NEED NOT PRESERVE POSITIVE RICCI CURVATURE CURTIS PRO AND FREDERICK WILHELM Abstract. If π : M B is a Riemannian Submersion and M has positive sectional curvature, O Neill s Horizontal
More informationEinstein H-umbilical submanifolds with parallel mean curvatures in complex space forms
Proceedings of The Eighth International Workshop on Diff. Geom. 8(2004) 73-79 Einstein H-umbilical submanifolds with parallel mean curvatures in complex space forms Setsuo Nagai Department of Mathematics,
More informationAN OBATA-TYPE THEOREM ON A THREE-DIMENSIONAL CR MANIFOLD
AN OBATA-TYPE THEORE ON A THREE-DIENSIONAL CR ANIFOLD S. IVANOV AND D. VASSILEV Abstract. We prove a CR version of the Obata s result for the first eigenvalue of the sub-laplacian in the setting of a compact
More informationLegendre surfaces whose mean curvature vectors are eigenvectors of the Laplace operator
Note di Matematica 22, n. 1, 2003, 9 58. Legendre surfaces whose mean curvature vectors are eigenvectors of the Laplace operator Tooru Sasahara Department of Mathematics, Hokkaido University, Sapporo 060-0810,
More informationCENTROAFFINE HYPEROVALOIDS WITH EINSTEIN METRIC
CENTROAFFINE HYPEROVALOIDS WITH EINSTEIN METRIC Udo Simon November 2015, Granada We study centroaffine hyperovaloids with centroaffine Einstein metric. We prove: the hyperovaloid must be a hyperellipsoid..
More informationEigenvalues of Collapsing Domains and Drift Laplacian
Eigenvalues of Collapsing Domains and Drift Laplacian Zhiqin Lu Dedicate to Professor Peter Li on his 60th Birthday Department of Mathematics, UC Irvine, Irvine CA 92697 January 17, 2012 Zhiqin Lu, Dept.
More informationf-biharmonic AND BI-f-HARMONIC SUBMANIFOLDS OF PRODUCT SPACES
SARAJEVO JOURNAL OF MATHEMATICS Vol.13 (5), No.1, (017), 115 19 DOI: 10.5644/SJM.13.1.09 f-biharmonic AND BI-f-HARMONIC SUBMANIFOLDS OF PRODUCT SPACES FATMA KARACA AND CIHAN ÖZGÜR Abstract. We consider
More informationThe Fundamental Gap. Mark Ashbaugh. May 19, 2006
The Fundamental Gap Mark Ashbaugh May 19, 2006 The second main problem to be focused on at the workshop is what we ll refer to as the Fundamental Gap Problem, which is the problem of finding a sharp lower
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationOn the spectrum of the Hodge Laplacian and the John ellipsoid
Banff, July 203 On the spectrum of the Hodge Laplacian and the John ellipsoid Alessandro Savo, Sapienza Università di Roma We give upper and lower bounds for the first eigenvalue of the Hodge Laplacian
More informationON THE CONSTRUCTION OF DUALLY FLAT FINSLER METRICS
Huang, L., Liu, H. and Mo, X. Osaka J. Math. 52 (2015), 377 391 ON THE CONSTRUCTION OF DUALLY FLAT FINSLER METRICS LIBING HUANG, HUAIFU LIU and XIAOHUAN MO (Received April 15, 2013, revised November 14,
More informationON THE FOLIATION OF SPACE-TIME BY CONSTANT MEAN CURVATURE HYPERSURFACES
ON THE FOLIATION OF SPACE-TIME BY CONSTANT MEAN CURVATURE HYPERSURFACES CLAUS GERHARDT Abstract. We prove that the mean curvature τ of the slices given by a constant mean curvature foliation can be used
More informationSome Remarks on Ricci Solitons
University of New Haven Digital Commons @ New Haven Mathematics Faculty Publications Mathematics 12-2017 Some Remarks on Ricci Solitons Ramesh Sharma University of New Haven, rsharma@newhaven.edu S Balasubramanian
More informationDefinition and basic properties of heat kernels I, An introduction
Definition and basic properties of heat kernels I, An introduction Zhiqin Lu, Department of Mathematics, UC Irvine, Irvine CA 92697 April 23, 2010 In this lecture, we will answer the following questions:
More informationLECTURE 21: THE HESSIAN, LAPLACE AND TOPOROGOV COMPARISON THEOREMS. 1. The Hessian Comparison Theorem. We recall from last lecture that
LECTURE 21: THE HESSIAN, LAPLACE AND TOPOROGOV COMPARISON THEOREMS We recall from last lecture that 1. The Hessian Comparison Theorem K t) = min{kπ γt) ) γt) Π γt) }, K + t) = max{k Π γt) ) γt) Π γt) }.
More informationTHE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS
THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily
More informationPseudo-Poincaré Inequalities and Applications to Sobolev Inequalities
Pseudo-Poincaré Inequalities and Applications to Sobolev Inequalities Laurent Saloff-Coste Abstract Most smoothing procedures are via averaging. Pseudo-Poincaré inequalities give a basic L p -norm control
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J. Math. Anal. 2 (2008), no., 70 77 Banach Journal of Mathematical Analysis ISSN: 735-8787 (electronic) http://www.math-analysis.org WIDTH-INTEGRALS AND AFFINE SURFACE AREA OF CONVEX BODIES WING-SUM
More informationChanging sign solutions for the CR-Yamabe equation
Changing sign solutions for the CR-Yamabe equation Ali Maalaoui (1) & Vittorio Martino (2) Abstract In this paper we prove that the CR-Yamabe equation on the Heisenberg group has infinitely many changing
More informationAnais da Academia Brasileira de Ciências ISSN: Academia Brasileira de Ciências Brasil
Anais da Academia Brasileira de Ciências ISSN: 0001-3765 aabc@abc.org.br Academia Brasileira de Ciências Brasil Ma, Li; Yang, Yang A remark on soliton equation of mean curvature flow Anais da Academia
More informationCOMPARISON THEOREMS FOR THE SPECTRAL GAP OF DIFFUSIONS PROCESSES AND SCHRÖDINGER OPERATORS ON AN INTERVAL. Ross G. Pinsky
COMPARISON THEOREMS FOR THE SPECTRAL GAP OF DIFFUSIONS PROCESSES AND SCHRÖDINGER OPERATORS ON AN INTERVAL Ross G. Pinsky Department of Mathematics Technion-Israel Institute of Technology Haifa, 32000 Israel
More informationSymplectic critical surfaces in Kähler surfaces
Symplectic critical surfaces in Kähler surfaces Jiayu Li ( Joint work with X. Han) ICTP-UNESCO and AMSS-CAS November, 2008 Symplectic surfaces Let M be a compact Kähler surface, let ω be the Kähler form.
More informationCMS winter meeting 2008, Ottawa. The heat kernel on connected sums
CMS winter meeting 2008, Ottawa The heat kernel on connected sums Laurent Saloff-Coste (Cornell University) December 8 2008 p(t, x, y) = The heat kernel ) 1 x y 2 exp ( (4πt) n/2 4t The heat kernel is:
More informationEigenvalue (mis)behavior on manifolds
Bucknell University Lehigh University October 20, 2010 Outline 1 Isoperimetric inequalities 2 3 4 A little history Rayleigh quotients The Original Isoperimetric Inequality The Problem of Queen Dido: maximize
More informationarxiv: v1 [math.dg] 21 Jun 2013
COMPARISON THEOREMS FOR MANIFOLD WITH MEAN CONVEX BOUNDARY arxiv:136.579v1 [math.dg] 21 Jun 213 JIAN GE Abstract. Let M n be an n-dimensional Riemannian manifold with boundary M. Assume that Ricci curvature
More informationPolyharmonic Elliptic Problem on Eistein Manifold Involving GJMS Operator
Journal of Applied Mathematics and Computation (JAMC), 2018, 2(11), 513-524 http://www.hillpublisher.org/journal/jamc ISSN Online:2576-0645 ISSN Print:2576-0653 Existence and Multiplicity of Solutions
More informationarxiv: v1 [math.mg] 28 Sep 2017
Ricci tensor on smooth metric measure space with boundary Bang-Xian Han October 2, 2017 arxiv:1709.10143v1 [math.mg] 28 Sep 2017 Abstract Theaim of this note is to studythemeasure-valued Ricci tensor on
More informationSCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze I. M. SINGER BUN WONG SHING-TUNG YAU STEPHEN S.-T. YAU An estimate of the gap of the first two eigenvalues in the Schrödinger operator Annali
More informationtheorem for harmonic diffeomorphisms. Theorem. Let n be a complete manifold with Ricci 0, and let n be a simplyconnected manifold with nonpositive sec
on-existence of Some Quasi-conformal Harmonic Diffeomorphisms Lei i Λ Department of athematics University of California, Irvine Irvine, CA 92697 lni@math.uci.edu October 5 997 Introduction The property
More informationInterpolating the arithmetic geometric mean inequality and its operator version
Linear Algebra and its Applications 413 (006) 355 363 www.elsevier.com/locate/laa Interpolating the arithmetic geometric mean inequality and its operator version Rajendra Bhatia Indian Statistical Institute,
More informationRiemannian Curvature Functionals: Lecture I
Riemannian Curvature Functionals: Lecture I Jeff Viaclovsky Park City athematics Institute July 16, 2013 Overview of lectures The goal of these lectures is to gain an understanding of critical points of
More informationA Characterization of Einstein Manifolds
Int. J. Contemp. Math. Sciences, Vol. 7, 212, no. 32, 1591-1595 A Characterization of Einstein Manifolds Simão Stelmastchuk Universidade Estadual do Paraná União da Vitória, Paraná, Brasil, CEP: 846- simnaos@gmail.com
More informationPENGFEI GUAN AND XI SISI SHEN. Dedicated to Professor D. Phong on the occasion of his 60th birthday
A RIGIDITY THEORE FOR HYPERSURFACES IN HIGHER DIENSIONAL SPACE FORS PENGFEI GUAN AND XI SISI SHEN Dedicated to Professor D. Phong on the occasion of his 60th birthday Abstract. We prove a rigidity theorem
More informationarxiv: v1 [math.co] 27 Jul 2012
Harnack inequalities for graphs with non-negative Ricci curvature arxiv:1207.6612v1 [math.co] 27 Jul 2012 Fan Chung University of California, San Diego S.-T. Yau Harvard University May 2, 2014 Abstract
More informationAFFINE SPHERES AND KÄHLER-EINSTEIN METRICS. metric makes sense under projective coordinate changes. See e.g. [10]. Form a cone (1) C = s>0
AFFINE SPHERES AND KÄHLER-EINSTEIN METRICS JOHN C. LOFTIN 1. Introduction In this note, we introduce a straightforward correspondence between some natural affine Kähler metrics on convex cones and natural
More informationarxiv: v1 [math.ap] 18 May 2017
Littlewood-Paley-Stein functions for Schrödinger operators arxiv:175.6794v1 [math.ap] 18 May 217 El Maati Ouhabaz Dedicated to the memory of Abdelghani Bellouquid (2/2/1966 8/31/215) Abstract We study
More informationDIAMETER, VOLUME, AND TOPOLOGY FOR POSITIVE RICCI CURVATURE
J. DIFFERENTIAL GEOMETRY 33(1991) 743-747 DIAMETER, VOLUME, AND TOPOLOGY FOR POSITIVE RICCI CURVATURE J.-H. ESCHENBURG Dedicated to Wilhelm Klingenberg on the occasion of his 65th birthday 1. Introduction
More informationRICCI SOLITONS ON COMPACT KAHLER SURFACES. Thomas Ivey
RICCI SOLITONS ON COMPACT KAHLER SURFACES Thomas Ivey Abstract. We classify the Kähler metrics on compact manifolds of complex dimension two that are solitons for the constant-volume Ricci flow, assuming
More informationA compactness theorem for Yamabe metrics
A compactness theorem for Yamabe metrics Heather acbeth November 6, 2012 A well-known corollary of Aubin s work on the Yamabe problem [Aub76a] is the fact that, in a conformal class other than the conformal
More informationON FRACTAL DIMENSION OF INVARIANT SETS
ON FRACTAL DIMENSION OF INVARIANT SETS R. MIRZAIE We give an upper bound for the box dimension of an invariant set of a differentiable function f : U M. Here U is an open subset of a Riemannian manifold
More informationGeometry and the Kato square root problem
Geometry and the Kato square root problem Lashi Bandara Centre for Mathematics and its Applications Australian National University 7 June 2013 Geometric Analysis Seminar University of Wollongong Lashi
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics CURVATURE CHARACTERIZATION OF CERTAIN BOUNDED DOMAINS OF HOLOMORPHY FANGYANG ZHENG Volume 163 No. 1 March 1994 PACIFIC JOURNAL OF MATHEMATICS Vol. 163, No. 1, 1994 CURVATURE
More informationNOTE ON THE SKEW ENERGY OF ORIENTED GRAPHS. Communicated by Ivan Gutman. 1. Introduction
Transactions on Combinatorics ISSN (print): 2251-8657, ISSN (on-line): 2251-8665 Vol. 4 No. 1 (2015), pp. 57-61. c 2015 University of Isfahan www.combinatorics.ir www.ui.ac.ir NOTE ON THE SKEW ENERGY OF
More informationBERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS
BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS SHOO SETO Abstract. These are the notes to an expository talk I plan to give at MGSC on Kähler Geometry aimed for beginning graduate students in hopes to motivate
More informationMARKOV PARTITIONS FOR HYPERBOLIC SETS
MARKOV PARTITIONS FOR HYPERBOLIC SETS TODD FISHER, HIMAL RATHNAKUMARA Abstract. We show that if f is a diffeomorphism of a manifold to itself, Λ is a mixing (or transitive) hyperbolic set, and V is a neighborhood
More informationANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS INVOLVING A NEW FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL
Electronic Journal of Differential Equations, Vol. 217 (217), No. 289, pp. 1 6. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS
More information