Bernstein theorems for length and area decreasing minimal maps

@article{SavasHalilaj2014BernsteinTF,
  title={Bernstein theorems for length and area decreasing minimal maps},
  author={Andreas Savas-Halilaj and Knut Smoczyk},
  journal={Calculus of Variations and Partial Differential Equations},
  year={2014},
  volume={50},
  pages={549-577}
}
In this article we prove Liouville and Bernstein theorems in higher codimension for length and area decreasing maps between two Riemannian manifolds. The proofs are based on a strong elliptic maximum principle for sections in vector bundles, which we also present in this article. 
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19 Citations
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Methods Citations
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19 Citations

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Graphical Mean Curvature Flow
  • A. Savas-Halilaj
  • Mathematics
    Nonlinear Analysis, Differential Equations, and Applications
  • 2021
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References

SHOWING 1-10 OF 47 REFERENCES
MINIMAL GRAPHS IN R WITH BOUNDED JACOBIANS
In this paper we obtain a Bernstein type result for entire two dimensional minimal graphs in R, which extends a previous one due to L. Ni. Moreover, we provide a characterization for complex analytic…
Bernstein type theorems with flat normal bundle
We prove Bernstein type theorems for minimal n-submanifolds in ℝn+p with flat normal bundle. Those are natural generalizations of the corresponding results of Ecker-Huisken and Schoen-Simon-Yau for…
Mean curvature flow of the graphs of maps between compact manifolds
We make several improvements on the results of M.-T. Wang (2002) and his joint paper with M.-P. Tsui (2004) concerning the long time existence and convergence for solutions of mean curvature flow in…
On graphic Bernstein type results in higher codimension
Let E be a minimal submanifold of R n+m that can be represented as the graph of a smooth map f: R n → R m . We apply a formula that we derived in the study of mean curvature flow to obtain conditions…
The regularity of harmonic maps into spheres and applications to Bernstein problems
We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of…
Mean curvature flows and isotopy of maps between spheres
Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A…
Mean Curvature Flows of Lagrangian Submanifolds with Convex Potentials
This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian…
Homotopy of area decreasing maps by mean curvature flow
The Ricci Flow in Riemannian Geometry
This book focuses on Hamilton’s Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness…
Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension
Abstract.Let f:Σ1↦Σ2 be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in Σ1×Σ2 by the mean curvature flow. Under suitable…
...
...