The smooth Riemannian extension problem
Stefano Pigola,Giona Veronelli +1 more
TLDR
In this paper, it was shown that it is always possible to realize a geodesically complete Riemannian extension without curvature constraints, even in the presence of a convexity condition on the boundary.Abstract:
Given a metrically complete Riemannian manifold $(M,g)$ with smooth nonempty boundary and assuming that one of its curvatures is subject to a certain bound, we address the problem of whether it is possibile to realize $(M,g)$ as a domain inside a geodesically complete Riemannian manifold $(M',g')$ without boundary, by preserving the same curvature bounds. In this direction we provide three kind of results: (1) a general existence theorem showing that it is always possible to obtain a geodesically complete Riemannian extension without curvature constraints; (2) various topological obstructions to the existence of a complete Riemannian extension with prescribed sectional and Ricci curvature bounds; (3) some existence results of complete Riemannian extensions with sectional and Ricci curvature bounds, mostly in the presence of a convexity condition on the boundary.read more
Citations
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Min-max theory for free boundary minimal hypersurfaces I - regularity theory
Martin Li,Xin Zhou +1 more
TL;DR: In this paper, the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain was proved for Riemannian manifolds with nonnegative Ricci curvature and convex boundary.
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The Steklov and Laplacian spectra of Riemannian manifolds with boundary
TL;DR: In this article, the existence of a constant C such that | σ k − λ k | ≤ C for each k ∈ N is proved. And the constant C depends only on the geometry of Ω 1 ≅ Ω 2.
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The Steklov and Laplacian spectra of Riemannian manifolds with boundary
TL;DR: In this article, the existence of a constant constant which depends only on the geometry of the Euclidean neighborhood of the Laplacian on the boundary of a compact Riemannian manifold is proved.
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Global Calder\'on-Zygmund inequalities on complete Riemannian manifolds
TL;DR: A survey of recent results on the validity and the failure of global regularity properties of smooth solutions of the Poisson equation on a complete Riemannian manifold can be found in this article.
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Dirichlet parabolicity and L1-Liouville property under localized geometric conditions
TL;DR: In this paper, a new light on the L 1 -Liouville property for positive, superharmonic functions is shed by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space.
References
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Book
A Course in Metric Geometry
TL;DR: In this article, a large-scale Geometry Spaces of Curvature Bounded Above Spaces of Bounded Curvatures Bounded Below Bibliography Index is presented. But it is based on the Riemannian metric space.
Book
Metric Structures for Riemannian and Non-Riemannian Spaces
TL;DR: In this paper, Loewner proposed a metric structure with a bounded Ricci Curvature for length structures on families of metric spaces, where the degree and dilatation of the length structure is a function of degree and degree.
Book
A First Course in Sobolev Spaces
TL;DR: Leoni as discussed by the authors takes a novel approach to the theory by looking at Sobolev spaces as the natural development of monotone, absolutely continuous, and BV functions of one variable.
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