The Convexity and the Gaussian Curvature Estimates for the Level Sets of Harmonic Functions on Convex Rings in Space Forms

In this paper, we first establish a constant rank theorem for the second fundamental form of the convex level sets of harmonic functions in space forms. Applying the deformation process, we prove that the level sets of the harmonic functions on convex rings in space forms are strictly convex. Moreover, we give a lower bound for the Gaussian curvature of the convex level sets of harmonic functions in terms of the Gaussian curvature of the boundary and the norm of the gradient on the boundary.     

Vanishing Theorems for Conformally Compact Manifolds

Abstract

We study L ² harmonic p-forms on conformally compact manifolds with a rather weak boundary regularity assumption. We proved that if the lower bound of the curvature operator is great than or equal to ?1 and the infimum of the L ² spectrum of the Laplacian great than p(n ? p) for some p ≤ n/2, then there is no nontrivial L ² harmonic p-form.     

Gaussian Curvature Estimates for the Convex Level Sets of Solutions for Some Nonlinear Elliptic Partial Differential Equations

We give lower bound estimates for the Gaussian curvature of convex level sets of minimal surfaces and the solutions to semilinear elliptic equations in terms of the norm of boundary gradient and the Gaussian curvature of the boundary.     

Gap Phenomena for p-Harmonic Maps

In this paper, we generalize to p-harmonic mapssome gap results known for harmonic maps. In particular, we prove that,under a certain level of energy depending on the curvature of the domainand target manifolds, the only p-harmonic maps are theconstant ones. The main tools are Bochner–Weitzenböck andReilly-type formulas involving the p-Laplace operator.     

Compact Surfaces with Constant Gaussian Curvature in Product Spaces

We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.     

Some sufficient conditions for immersion in the large of metrics of positive curvature

It is proved that a regular Riemannian manifold diffeomorphic to a circle and having positive Gaussian curvature bounded from zero is immersible into a three-dimensional Euclidean space in the form of a regular surface if it has smallL _p (the norm of the gradient of Gaussian curvature), p > 2, or if it has a sufficiently small area (with any behavior of the geodesic boundary curvature).Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 51–59, 1991.     

Holomorphic two-spheres in complex Grassmann manifold <Emphasis Type="Italic">G</Emphasis>(2, 4)

In this paper, we use the harmonic sequence to study the linearly full holomorphic two-spheres in complex Grassmann manifold G(2, 4). We show that if the Gaussian curvature K (with respect to the induced metric) of a non-degenerate holomorphic two-sphere satisfies K ≤ 2 (or K ≥ 2), then K must be equal to 2. Simultaneously, we show that one class of the holomorphic two-spheres with constant curvature 2 is totally geodesic. Concerning the degenerate holomorphic two-spheres, if its Gaussian curvature K ≤ 1 (or K ≥ 1), then K = 1. Moreover, we prove that all holomorphic two-spheres with constant curvature 1 in G(2, 4) must be U (4)-equivalent.     

Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball

A representation of the sharp constant in a pointwise estimate for the absolute value of the directional derivative of a harmonic function in a multidimensional ball is obtained under the assumption that the boundary values of the function belong to L ^p . This representation is specified in the cases of radial and tangential derivatives. It is proved for p = 1 and p = 2 that the maximum of the absolute value of the directional derivative of a harmonic function with a fixed L ^p -norm of its boundary values is attained at the radial direction. This confirms D. Khavinson’s conjecture for p = 1 and p = 2. Bibliography: 11 titles.     

Generalized motion of level sets by functions of their curvatures on Riemannian manifolds

We consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms on a Riemannian manifold M. The level sets of a function evolve in such a way whenever u solves an equation u _t  + F(Du, D ² u) = 0, for some real function F satisfying a geometric condition. We show existence and uniqueness of viscosity solutions to this equation under the assumptions that M has nonnegative curvature, F is continuous off {Du = 0}, (degenerate) elliptic, and locally invariant by parallel translation. We then prove that this approach is geometrically consistent, hence it allows to define a generalized evolution of level sets by very general, singular functions of their curvatures. For instance, these assumptions on F are satisfied when F is given by the evolutions of level sets by their mean curvature (even in arbitrary codimension) or by their positive Gaussian curvature. We also prove that the generalized evolution is consistent with the classical motion by the corresponding function of the curvature, whenever the latter exists. When M is not of nonnegative curvature, the same results hold if one additionally requires that F is uniformly continuous with respect to D ² u. Finally we give some counterexamples showing that several well known properties of the evolutions in are no longer true when M has negative sectional curvature. D. Azagra was supported by grants MTM-2006-03531 and UCM-CAM-910626. M. Jimenez-Sevilla was supported by a fellowship of the Ministerio de Educacion y Ciencia, Spain. F. Macià was supported by program “Juan de la Cierva” and projects MAT2005-05730-C02-02 of MEC (Spain) and PR27/05-13939 UCM-BSCH (Spain).     

Equiaffine geometry of level sets and ruled hypersurfaces with equiaffine mean curvature zero

Basic aspects of the equiaffine geometry of level sets are developed systematically. As an application there are constructed families of 2n‐dimensional nondegenerate hypersurfaces ruled by n‐planes, having equiaffine mean curvature zero, and solving the affine normal flow. Each carries a symplectic structure with respect to which the ruling is Lagrangian.     

The Willmore conjecture for immersed tori with small curvature integral

The Willmore conjecture states that any immersion of a 2-torus into euclidean space satisfies . We prove it under the condition that the L ^p -norm of the Gaussian curvature is sufficiently small. Received: 10 June 1999     

Existence of solution in elastic wave scattering by unbounded rough surfaces

We consider the two‐dimensional problem of the scattering of a time‐harmonic wave, propagating in an homogeneous, isotropic elastic medium, by a rough surface on which the displacement is assumed to vanish. This surface is assumed to be given as the graph of a function ?∈C^1,1(?). Following up on earlier work establishing uniqueness of solution to this problem, existence of solution is studied via the boundary integral equation method. This requires a novel approach to the study of solvability of integral equations on the real line. The paper establishes the existence of a unique solution to the boundary integral equation formulation in the space of bounded and continuous functions as well as in all L^p spaces, p∈[1, ∞] and hence existence of solution to the elastic wave scattering problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains

In this paper we prove new results for p harmonic functions, p≠2, 1<p<∞, in Lipschitz and starlike Lipschitz ring domains. In particular we prove the boundary Harnack inequality, Theorem 1, for the ratio of two positive p harmonic functions vanishing on a portion of the boundary of a Lipschitz domain, with constants only depending on p,n and the Lipschitz constant of the domain. For p capacitary functions, in starlike Lipschitz ring domains, we prove an even stronger result, Theorem 2, showing that the ratio is Hölder continuous up to the boundary. Moreover, for p capacitary functions in starlike Lipschitz ring domains we prove, Theorems 3 and 4, appropriate extensions to p≠2, 1<p<∞, of famous results of Dahlberg [12] and Jerison and Kenig [25] on the Poisson kernel associated to the Laplace operator (i.e. p=2).     

Generalized bochner theorems and the spectrum of complete manifolds

Bochner's theorem that a compact Riemannian manifold with positive Ricci curvature has vanishing first cohomology group has various extensions to complete noncompact manifolds with Ricci possibly negative. One still has a vanishing theorem for L ² harmonic one-forms if the infimum of the spectrum of the Laplacian on functions is greater than minus the infimum of the Ricci curvature. This result and its analogues for p-forms yield vanishing results for certain infinite volume hyperbolic manifolds. This spectral condition also imposes topological restrictions on the ends of the manifold. More refined results are obtained by taking a certain Brownian motion average of the Ricci curvature; if this average is positive, one has a vanishing theorem for the first cohomology group with compact supports on the universal cover of a compact manifold. There are corresponding results for L ² harmonic spinors on spin manifolds.     

Onp-harmonic functions in the complex plane and curvature

In the complex plane thep-harmonic equation div(|∇u| ^p−2∇u) = 0, 1 <p < ∞, exhibits some features reminiscent of Function Theory. Our results about curvature in this structure complement known facts about minimal surfaces and harmonic functions. Quasiregular mappings are used.     

Existence of non-trivial harmonic functions on Cartan–Hadamard manifolds of unbounded curvature

The Liouville property of a complete Riemannian manifold M (i.e., the question whether there exist non-trivial bounded harmonic functions on M) attracted a lot of attention. For Cartan–Hadamard manifolds the role of lower curvature bounds is still an open problem. We discuss examples of Cartan–Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan–Hadamard manifolds is much more complicated than one might have expected.      

Thin sets of integers in harmonic analysis and <Emphasis Type="Italic">p</Emphasis>-stable random fourier series

We investigate the behaviour of some thin sets of integers defined through random trigonometric polynomials when one replaces Gaussian or Rademacher variables with p-stable ones, 1 < p < 2. We show that in one case, this behaviour is essentially the same as in the Gaussian case, whereas in another case, it is entirely different.     

Space and time inversions of stochastic processes and Kelvin transform

Let X be a standard Markov process. We prove that a space inversion property of X implies the existence of a Kelvin transform of X‐harmonic, excessive and operator‐harmonic functions and that the inversion property is inherited by Doob h‐transforms. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. We treat in details the examples of free scaled power Bessel processes, non‐colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.     

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.     

Local Symmetry of Harmonic Spaces as Determined by The Spectra of Small Geodesic Spheres

We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm |?R|{{|\nabla{R}|}} of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determine whether the space is locally symmetric. For the proof we use the first few heat invariants and consider certain coefficients in the radial power series expansions of the curvature invariants |R|² and |Ric|² of the geodesic spheres. Moreover, we obtain analogous results for geodesic balls with either Dirichlet or Neumann boundary conditions. We also comment on the relevance of these results to constructions of Z.I. Szabó.