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In this paper, we prove two Liouville theorems for harmonic maps and apply them to study the topology of manifolds with positive spectrum and stable minimal hypersurfaces in Riemannian manifolds with non-negative bi-Ricci curvature.

Harmonic Maps and the Topology of Stable Hypersurfaces and Manifolds with Non-negative Ricci Curvature

Shing Tung Yau, Richard Schoen (1976)

Commentarii mathematici Helvetici

Harmonic maps from a Riemannian manifold with a pole into an Hadamard manifold with negative sectional curvatures.

Atsushi Tachikawa (1992)

Manuscripta mathematica

Harmonic maps on spaces with conical singularities

Y.-J. Chiang, A. Ratto (1992)

Bulletin de la Société Mathématique de France

Harmonic morphisms between riemannian manifolds

Bent Fuglede (1978)

Annales de l'institut Fourier

A harmonic morphism $f : M \to N$ between Riemannian manifolds $M$ and $N$ is by definition a continuous mappings which pulls back harmonic functions. It is assumed that dim $M \geq$ dim $N$ , since otherwise every harmonic morphism is constant. It is shown that a harmonic morphism is the same as a harmonic mapping in the sense of Eells and Sampson with the further property of being semiconformal, that is, a conformal submersion of the points where $d f$ vanishes. Every non-constant harmonic morphism is shown to be an open mapping....

Harmonic $ϕ$ -morphisms.

Bejan, C. L., Benyounes, M. (2003)

Beiträge zur Algebra und Geometrie

Harmonie reflections

Lieven Vanhecke, Maria-Elena Vazquez-Abal (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We study local reflections $\phi_{\sigma}$ with respect to a curve $\sigma$ in a Riemannian manifold and prove that $\sigma$ is a geodesic if $\phi_{\sigma}$ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if $\phi_{\sigma}$ is harmonic for all geodesies $\sigma$ .

Harnack inequalities for curvature flows depending on mean curvature.

Smoczyk, Knut (1997)

The New York Journal of Mathematics [electronic only]

Heat flow of p-harmonic maps with values into spheres.

Y. Chen, M.-C. Hong, N. Hungerbühler (1994)

Mathematische Zeitschrift

Heat kernel estimates and lower bound of eigenvalues.

Shiu-Yuen Cheng, Peter Li (1981)

Commentarii mathematici Helvetici

Heterogeneous Riemannian manifolds.

Hebda, James J. (2010)

International Journal of Mathematics and Mathematical Sciences

Higher order Codazzi tensors on conformally flat spaces.

Liu, H.L., Simon, U., Wang, C.P. (1998)

Beiträge zur Algebra und Geometrie

High-order angles in almost-Riemannian geometry

Ugo Boscain, Mario Sigalotti (2006/2007)

Séminaire de théorie spectrale et géométrie

Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$ . If $X$ and $Y$ are everywhere linearly independent, then they define a Riemannian metric on $M$ (the metric for which they are orthonormal) and they give to $M$ the structure of metric space. If $X$ and $Y$ become linearly dependent somewhere on $M$ , then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way...

Histoire de la réductibilité du groupe conforme des variétés riemanniennes (1964-1994)

Jacqueline Ferrand (1998/1999)

Séminaire de théorie spectrale et géométrie

Holonomy Groups of a Fully Parallelizable Manifold

Jiří Vanžura (1970)

Matematický časopis

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