Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds

Dini derivatives in Riemannian manifold settings are studied in this paper. In addition, a characterization for Lipschitz and convex functions defined on Riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the Dini derivative are given.     

Harmonic functions on Riemannian manifolds with ends

We study the problem of solvability of some boundary value problems on noncompact Riemannian manifolds with ends. We obtain the conditions for existence and uniqueness of solutions to the problems as well as the conditions for the fulfillment of Liouville-type theorems for harmonic functions on the manifolds.     

Proximal subgradient and a characterization of Lipschitz function on Riemannian manifolds

A characterization of Lipschitz behavior of functions defined on Riemannian manifolds is given in this paper. First, it is extended the concept of proximal subgradient and some results of proximal analysis from Hilbert space to Riemannian manifold setting. A technique introduced by Clarke, Stern and Wolenski [F.H. Clarke, R.J. Stern, P.R. Wolenski, Subgradient criteria for monotonicity, the Lipschitz condition, and convexity, Canad. J. Math. 45 (1993) 1167-1183], for generating proximal subgradients of functions defined on a Hilbert spaces, is also extended to Riemannian manifolds in order to provide that characterization. A number of examples of Lipschitz functions are presented so as to show that the Lipschitz behavior of functions defined on Riemannian manifolds depends on the Riemannian metric.     

Generalized distance functions of Riemannian manifolds and the motions of gyroscopic systems

We use the homology groups of the path space of an arbitrary Riemannian manifold to define some analogs of the distance function and study their main properties. For the natural systems with gyroscopic forces we prove an existence theorem for solutions to the two-point boundary value problem, which complements the results of [1]. We apply the geodesic modeling method of [1, 2], using the generalized distance functions.     

Generalized Einstein conditions on holomorphic Riemannian manifolds

Summary At first, a necessary and sufficient condition for a K?hler-Norden manifold to be holomorphic Einstein is found. Next, it is shown that the so-called (real) generalized Einstein conditions for K?hler-Norden manifolds are not essential since the scalarcurvature of such manifolds is constant. In this context, we study generalized holomorphic Einstein conditions. Using the one-to-one correspondence between K?hler-Norden structures and holomorphic Riemannian metrics, we establish necessary and sufficient conditions for K?hler-Norden manifolds to satisfy the generalized holomorphic Einstein conditions. And a class of new examples of such manifolds is presented. Finally, in virtue of the obtained results, we mention that Theorems 1 and 2 of H. Kim and J. Kim [10] are not true in general.     

Existence and uniqueness for a semilinear elliptic problem on Lipschitz domains in Riemannian manifolds II

Extending our recent work for the semilinear elliptic equation on Lipschitz domains, we study a general second-order Dirichlet problem in . We improve our previous results by studying more general nonlinear terms with polynomial (and in some cases exponential) growth in the variable . We also study the case of nonnegative solutions.

     

Lipschitz precompactness for closed negatively curved manifolds

We prove that, given a integer and a group , the class of closed Riemannian -manifolds of uniformly bounded negative sectional curvatures and with fundamental groups isomorphic to is precompact in the Lipschitz topology. In particular, the class breaks into finitely many diffeomorphism types.

     

Pullback of measures on Riemannian manifolds

A general formula for the pullback of measures on submanifolds of Riemannian manifolds is derived. We prove the equivalence of three different approaches to distributions supported on submanifolds, originating from Gelfand and Shilov, Friedlander, and a characterization using single-layer distributions, respectively.     

Weighted diffeomorphism groups of Riemannian manifolds

In this paper, we define locally convex vector spaces of weighted vector fields and use them as model spaces for Lie groups of weighted diffeomorphisms on Riemannian manifolds. We prove an easy condition on the weights that ensures that these groups contain the compactly supported diffeomorphisms. We finally show that for the special case where the manifold is the euclidean space, these Lie groups coincide with the ones constructed in the author’s earlier work (Walter, 2012).     

Existence of solutions for variational inequalities on Riemannian manifolds

We establish the existence and uniqueness results for variational inequality problems on Riemannian manifolds and solve completely the open problem proposed in [S.Z. Németh, Variational inequalities on Hadamard manifolds, Nonlinear Anal. 52 (2003) 1491–1498]. Also the relationships between the constrained optimization problem and the variational inequality problems as well as the projections on Riemannian manifolds are studied.     

Approximation of a point perturbation on a Riemannian manifold

We show that the Hamiltonian of point interaction on a Riemannian manifold with bounded geometry can be obtained as a limit (in the sense of uniform resolvent convergence) of a sequence of scaling Hamiltonians with short-range interaction. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 1, pp. 49–57, January, 2009.     

Harmonic Riemannian maps on locally conformal Kaehler manifolds

We study harmonic Riemannian maps on locally conformal Kaehler manifolds (lcK manifolds). We show that if a Riemannian holomorphic map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the lcK manifold is Kaehler. Then we find similar results for Riemannian maps between lcK manifolds and Sasakian manifolds. Finally, we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.     

Eigenvalue estimate on complete noncompact Riemannian manifolds and applications

We obtain some sharp estimates on the first eigenvalues of complete noncompact Riemannian manifolds under assumptions of volume growth. Using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures.

     

Maximal complexifications of certain homogeneous Riemannian manifolds

Let be a homogeneous Riemannian manifold with , where denotes the universal complexification of . Under certain extensibility assumptions on the geodesic flow of , we give a characterization of the maximal domain of definition in for the adapted complex structure and show that it is unique. For instance, this can be done for generalized Heisenberg groups and naturally reductive homogeneous Riemannian spaces. As an application it is shown that the case of generalized Heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex.

     

Smooth approximation of definable continuous functions

Let be an -minimal expansion of the real exponential field which possesses smooth cell decomposition. We prove that for every definable open set, the definable indefinitely continuously differentiable functions are a dense subset of the definable continuous function with respect to the -minimal Whitney topology.

     

Killing vector fields of constant length on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on complete smooth Riemannian manifolds. Various examples are constructed of the Killing vector fields of constant length generated by the isometric effective almost free but not free actions of S ¹ on the Riemannian manifolds close in some sense to symmetric spaces. The latter manifolds include “almost round” odd-dimensional spheres and unit vector bundles over Riemannian manifolds. We obtain some curvature constraints on the Riemannian manifolds admitting nontrivial Killing fields of constant length.     

Heat content and Hardy inequality for complete Riemannian manifolds

Upper bounds are obtained for the heat content of an open set D in a complete Riemannian manifold, provided the Dirichlet-Laplace-Beltrami operator satisfies a strong Hardy inequality, and the distance function on D satisfies an integrability condition.     

Two realizations of Schrödinger operators on Riemannian manifolds

We consider a Schrödinger differential expression P=ΔM+V on a complete Riemannian manifold (M,g) with metric g, where ΔM is the scalar Laplacian on M and V is a real-valued locally integrable function on M. We study two self-adjoint realizations of P in L²(M) and show their equality. This is an extension of a result of S. Agmon.