Rough Isometry and <Emphasis Type="Italic">p</Emphasis>-Harmonic Boundaries of Complete Riemannian Manifolds

In this paper, we describe the behavior of bounded energy finite solutions for certain nonlinear elliptic operators on a complete Riemannian manifold in terms of its p-harmonic boundary. We also prove that if two complete Riemannian manifolds are roughly isometric to each other, then their p-harmonic boundaries are homeomorphic to each other. In the case, there is a one to one correspondence between the sets of bounded energy finite solutions on such manifolds. In particular, in the case of the Laplacian, it becomes a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral on the manifolds. This work was supported by grant No. R06-2002-012-01001-0(2002) from the Basic Research Program of the Korea Science & Engineering Foundation.     

A remark on convex functions andp-harmonic maps

We prove that the constant maps are the onlyp-harmonic maps for anyp 2 from an arbitrary compact Riemannian manifold into a complete Riemannian manifold which admits a strictly convex function.     

Existence and Liouville theorems for V -harmonic maps from complete manifolds

We establish existence and uniqueness theorems for V-harmonic maps from complete noncompact manifolds. This class of maps includes Hermitian harmonic maps, Weyl harmonic maps, affine harmonic maps, and Finsler harmonic maps from a Finsler manifold into a Riemannian manifold. We also obtain a Liouville type theorem for V-harmonic maps. In addition, we prove a V-Laplacian comparison theorem under the Bakry-Emery Ricci condition.     

L-harmonic functions with polynomial growth of a fixed rate

Yau made the following conjecture: For a complete noncompact manifold with nonnegative Ricci curvature the space of harmonic functions with polynomial growth of a fixed rate is finite dimensional. we extend the result on the Laplace operator to that on the symmetric diffusion operator, and prove the space of L-harmonic functions with polynomial growth of a fixed rate is finite-dimensional, when m-dimensional Bakery-Emery Ricci curvature of the symmetric diffusion operator on the complete noncompact Riemannian manifold is nonnegative.     

Complete harmonic stable minimal hypersurfaces in a Riemannian manifold

In this paper, we will introduce the notion of harmonic stability for complete minimal hypersurfaces in a complete Riemannian manifold. The first result we prove, is that a complete harmonic stable minimal surface in a Riemannian manifold with non-negative Ricci curvature is conformally equivalent to either a plane R ² or a cylinder R × S ¹, which generalizes a theorem due to Fischer-Colbrie and Schoen [12]. The second one is that an n ≥ 2-dimensional, complete harmonic stable minimal, hypersurface M in a complete Riemannian manifold with non-negative sectional curvature has only one end if M is non-parabolic. The third one, which we prove, is that there exist no non-trivial L ²-harmonic one forms on a complete harmonic stable minimal hypersurface in a complete Riemannian manifold with non-negative sectional curvature. Since the harmonic stability is weaker than stability, we obtain a generalization of a theorem due to Miyaoka [20] and Palmer [21]. Research partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The author’s research was supported by grant Proj. No. KRF-2007-313-C00058 from Korea Research Foundation, Korea. Authors’ addresses: Qing-Ming Cheng, Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan; Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea     

Classification and Liouville type theorems for <Emphasis Type="Italic">p</Emphasis>-harmonic morphisms

We prove firstly the classification theorem for p-harmonic morphisms between Euclidean domains. Secondly, we show that if is a p-harmonic morphism (p ≥ 2) from a complete Riemannian manifold M of nonnegative Ricci curvature into a Riemannian manifold N of non-positive scalar curvature such that the L ^q -energy is finite, then is constant, which improve the corresponding result due to G. Choi, G. Yun in (Geometriae Dedicata 101 (2003), 53–59).      

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L ² harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover, we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.     

Generalized Liouville property for Schrödinger operator on Riemannian manifolds

In this paper, we prove that the dimension of the space of positive (bounded, respectively) -harmonic functions on a complete Riemannian manifold with -regular ends is equal to the number of ends (-nonparabolic ends, respectively). This result is a solution of an open problem of Grigor'yan related to the Liouville property for the Schr?dinger operator . We also prove that if a given complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincaré inequality and the finite covering condition on each end, then the dimension of the space of positive (bounded, respectively,) solutions for the Schr?dinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends (-nonparabolic ends, respectively). This is a partial answer of the question, suggested by Li, related to the regularity of ends of a complete Riemannian manifold. Especially, our results directly generalize various earlier results of Yau, of Li and Tam, of Grigor'yan, and of present authors, but with different techniques that the peculiarity of the Schr?dinger operator demands. Received: 4 April 2000; in final form: 19 September 2000 / Published online: 25 June 2001     

Constancy of p-harmonic maps of finite q-energy into non-positively curved manifolds

We investigate p-harmonic maps, p ≥ 2, from a complete non-compact manifold into a non-positively curved target. First, we establish a uniqueness result for the p-harmonic representative in the homotopy class of a constant map. Next, we derive a Caccioppoli inequality for the energy density of a p-harmonic map and we prove a companion Liouville type theorem, provided the domain manifold supports a Sobolev–Poincaré inequality. Finally, we obtain energy estimates for a p-harmonic map converging, with a certain speed, to a given point.      

Indice local pour des ensembles symétriques par arcs

Let M be a complete Riemannian manifold with sectional curvature and dimension . Given a unit vector and a point we prove the existence of a complete geodesic through x whose tangent vector never comes close to v. As a consequence we show the existence of a bounded geodesic through every point in a complete negatively pinched manifold with finite volume and dimension . Received April 13, 1998; in final form July 23, 1999 / Published online October 11, 2000     

Stable ℱ-harmonic maps between Finsler manifolds

In this paper, we derive the first and second variation formulas for JC-harmonic maps between Finsler manifolds, and when F″≤ 0 and n ≥ 3, we prove that there is no nondegenerate stable F-harmonic map between a Riemannian unit sphere Sn and any compact Finsler manifold.     

The Oshima-Sekiguchi and Liouville theorems on Heintze groups

Let L be an elliptic operator on a Riemannian manifold M. A function F annihilated by L is said to be L-harmonic. F is said to have moderate growth if and only if F grows at most exponentially in the Riemannian distance. If M is a rank-one symmetric space and L is the Laplace-Beltrami operator for M, the Oshima-Sekiguchi theorem [T. Oshima, J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980) 1-81] states that a L-harmonic function F has moderate growth if and only if F is the Poisson integral of a distribution on the Furstenberg boundary. In this work we prove that this result generalizes to a very large class of homogeneous Riemannian manifolds of negative curvature. We also (i) prove a Liouville type theorem that characterizes the “polynomial-like” harmonic functions which vanish on the boundary in terms of their growth properties, (ii) describe all “polynomial-like” harmonic functions, and (iii) give asymptotic expansions for the Poisson kernel. One consequence of this work is that every Schwartz distribution on the boundary is the boundary value for a L-harmonic function F which is uniquely determined modulo “polynomial-like” harmonic functions.     

A Theorem of Liouville Type for p-Harmonic Morphisms

In this article we prove a Liouville type theorem for p-harmonic morphisms. We show that if : MNis a p-harmonic morphism (p2) from a complete noncompact Riemannian manifold Mof nonnegative Ricci curvature into a Riemannian manifold Nof nonpositive scalar curvature such that the p-energy E _p (), or (2p–2)-energy E _2p–2() is finite, then is constant.     

Affine symplectic geometry I: Applications to geometric inequalities

We use the integral geometric formulas in the symplectic space of geodesics of a Riemannian manifold to derive various inequalities of isoperimetric type. We give a sharp lower bound for the area of the minimal bubble spanning a spherical curve in ℝ³. We also present an “inverse Croke inequality” relating the area of the boundary of a complex domain in a Riemannian manifold to the injectivity radius and the volume of the domain. We prove a sharp lower bound for the ground state of the harmonic oscillator operator inL ²(M), whereM is a Hadamard manifold. This article is dedicated to my dear friend Julia Rashba     

Asymptotic Boundary Value Problem of Harmonic Maps via Harmonic Boundary

We prove that given any continuous data f on the harmonic boundary of a complete Riemannian manifold with image within a ball in the normal range, there exists a harmonic map from the manifold into the ball taking the same boundary value at each harmonic boundary point as that of f.     

The p-Harmonic Heat Flow with Potential into Homogeneous Spaces

In this paper we prove the existence of global weak solutions of the p-harmonic flow with potential between Riemannian manifolds M and N for arbitrary initial data having finite p-energy in the case when the target N is a homogeneous space with a left invariant metric. Received March 17, 1999, Revised September 22, 1999, Accepted October 15, 1999     

Nonnegative solutions of an elliptic equation and Harnack ends of Riemannian manifolds

In this paper, we give a barrier argument at infinity for solutions of an elliptic equation on a complete Riemannian manifold. By using the barrier argument, we can construct a nonnegative (bounded, respectively) solution of the elliptic equation, which takes the given data at infinity of each end. In particular, we prove that if a complete Riemannian manifold has finitely many ends, each of which is Harnack and nonparabolic, then the set of bounded solutions of the elliptic equation is finite dimensional, in some sense. We also prove that if a complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincaré inequality and the finite covering condition on each end, then there exists a nonnegative solution of an elliptic equation taking the given data at infinity of each end of the manifold. These results generalize those of Yau, of Donnelly, of Grigor'yan, of Li and Tam, of Holopainen, and of the present authors, but with the barrier argument at infinity that enables one to overcome the obstacle due to the nonlinearity of solutions. Received: 11 November 1999     

Flat Riemannian manifolds are boundaries

In this paper we prove that each compact flat Riemannian manifold is the boundary of a compact manifold. Our method of proof is to construct a smooth action of (₂) ^k on the flat manifold. We are independently preceded in this approach by Marc W. Gordon who proved the flat Riemannian manifolds, whose holonomy groups are of a certain class of groups, bound. By analyzing the fixed point data of this group action we get the complete result. As corollaries to the main theorem it follows that those compact flat Riemannian manifolds which are oriented bound oriented manifolds; and, if we have an involution on a homotopy flat manifold, then the manifold together with the involution bounds. We also give an example of a nonbounding manifold which is finitely covered byS ³ ×S ³ ×S ³.     

Random data Cauchy theory for supercritical wave equations I: local theory

We study the local existence of strong solutions for the cubic nonlinear wave equation with data in H ^s (M), s<1/2, where M is a three dimensional compact Riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in H ^s (M), where s≥1/4 in the case of a boundary less manifold and s≥8/21 in the case of a manifold with boundary. Mathematics Subject Classification (2000)  35Q55, 35BXX, 37K05, 37L50, 81Q20     

Bubbling phenomena of Palais-Smale-like sequences ofm-harmonic type systems

In this paper, we study the bubbling phenomena of weak solution sequences of a class of degenerate quasilinear elliptic systems ofm-harmonic type. We prove that, under appropriate conditions, the energy is preserved during the bubbling process. The results apply tom-harmonic maps from a closed Riemannian manifoldM to a Riemannian homogeneous space, and tom-harmonic maps with constant volumes, and also to certain Palais-Smale sequences.