Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative ricci curvature

@article{Shoen1976HarmonicMA,
  title={Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative ricci curvature},
  author={Richard L. Shoen and Shing-Tung Yau},
  journal={Commentarii Mathematici Helvetici},
  year={1976},
  volume={51},
  pages={333-341}
}
View on Springer
193 Citations
Highly Influential Citations
14
Background Citations
43
Methods Citations
15
Results Citations
4

193 Citations

Recent rigidity results for graphs with prescribed mean curvature
This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs u : M → $\mathbb{R}$. Emphasis is put on minimal, CMC and capillary…
Regularity and quantitative gradient estimate of p-harmonic mappings between Riemannian manifolds
Let $M$ be a $C^2$-smooth Riemannian manifold with boundary and $N$ a complete $C^2$-smooth Riemannian manifold. We show that each minimizing $p$-harmonic mapping $u\colon M\to N$ is locally…
Nonlinear partial differential systems on Riemannian manifolds with their geometric applications
We make the first study of how the existence of (essential) positive supersolutions of nonlinear degenerate partial differential equations on a manifold affects the topology, geometry, and analysis…
A Note on p-Harmonic 1-Forms on Complete Manifolds
  • Xi Zhang
  • Mathematics
    Canadian Mathematical Bulletin
  • 2001
Abstract In this paper we prove that there is no nontrivial ${{L}^{q}}$ -integrably $p$ -harmonic 1-form on a complete manifold with nonnegatively Ricci curvature $\left( 0\,<\,q\,<\,\infty \right)$ .
From Harmonic Mappings to Ricci Flows Due to the Bochner Technique
The present paper is devoted to the study a global aspect of the geometry of harmonic mappings and, in particular, infinitesimal harmonic transformations, and represents the application of our…
On Uniqueness Theorems for Ricci Tensor *
In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold M and a symmetric 2-tensor r, construct a metric on M whose Ricci tensor equals r. In particular,…
On DeTurck uniqueness theorems for Ricci tensor
In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In…
Topology of steady and expanding gradient Ricci solitons via f-harmonic maps
Vanishing theorems on complete manifolds with weighted Poincaré inequality and applications
Abstract Two vanishing theorems for harmonic map and L2 harmonic 1-form on complete noncompact manifolds are proved under certain geometric assumptions, which generalize results of [13], [15], [18],…
Stokes' theorem, volume growth and parabolicity
We present some new Stokes' type theorems on complete non-compact manifolds that extend, in different directions, previous work by Gaffney and Karp and also the so called Kelvin-Nevanlinna-Royden…
...
...

References

SHOWING 1-7 OF 7 REFERENCES
Harmonic Mappings of Riemannian Manifolds
On the Structure of Complete Manifolds of Nonnegative Curvature
Minimal submanifolds in a Riemannian manifold
The splitting theorem for manifolds of nonnegative Ricci curvature
On Complete Open Manifolds of Positive Curvature
Growth of finitely generated solvable groups and curvature of Riemannian manifolds
If a group Γ is generated by a finite subset 5, then one has the "growth function" gs, where gs(m) is the number of distinct elements of Γ expressible as words of length <m on 5. Roughly speaking, J.…
A note on curvature and fundamental group
Define the growth function γ associated with a finitely generated group and a specified choice of generators {gl7 -, gp} for the group as follows (compare [9]). For each positive integer s let γ(s)…