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Geometric singularities and a flow tangent to the Ricci flow
TLDR
In this article, the Ricci flow has been studied in the context of smooth manifolds with rough metrics with sufficiently regular heat kernels, and it is shown that the distance induced by the flow is identical to the evolving distance metric defined by Gigli and Mantegazza on appropriate admissible points.Abstract:
We consider a geometric flow introduced by Gigli and Mantegazza which, in the case of smooth compact manifolds with smooth metrics, is tangen- tial to the Ricci flow almost-everywhere along geodesics. To study spaces with geometric singularities, we consider this flow in the context of smooth manifolds with rough metrics with sufficiently regular heat kernels. On an appropriate non- singular open region, we provide a family of metric tensors evolving in time and provide a regularity theory for this flow in terms of the regularity of the heat kernel.
When the rough metric induces a metric measure space satisfying a Riemannian Curvature Dimension condition, we demonstrate that the distance induced by the flow is identical to the evolving distance metric defined by Gigli and Mantegazza on appropriate admissible points. Consequently, we demonstrate that a smooth compact manifold with a finite number of geometric conical singularities remains a smooth manifold with a smooth metric away from the cone points for all future times. Moreover, we show that the distance induced by the evolving metric tensor agrees with the flow of RCD(K, N) spaces defined by Gigli-Mantegazza.read more
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Continuity of solutions to space-varying pointwise linear elliptic equations
TL;DR: In this article, the authors consider pointwise linear elliptic equations of the form Lα uα = ŋα on a smooth compact manifold where the operators Lα are in divergence form with real, bounded, measurable coefficients that vary in the space variableα.
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On Weak Super Ricci Flow through Neckpinch
Sajjad Lakzian,Michael Munn +1 more
TL;DR: In this article, the Ricci flow neckpinch is studied in the context of metric measure spaces, and the concept of weak super Ricci flows associated with convex cost functions is introduced.
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Heat kernels and regularity for rough metrics on smooth manifolds
Lashi Bandara,Paul Bryan +1 more
TL;DR: In this paper, it was shown that globally continuous heat kernels exist and are H\"older continuous locally in space and time, via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.
References
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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.
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TL;DR: In this article, a curvature-dimension condition CD(K, N) for metric measure spaces is introduced, which is more restrictive than the curvature bound for Riemannian manifolds.
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Ricci curvature for metric-measure spaces via optimal transport
John Lott,Cédric Villani +1 more
TL;DR: In this paper, a notion of a length space X having nonnegative N-Ricci curvature, for N 2 [1;1], or having 1-RICci curvatures bounded below by K, for K2 R, was given.