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Width estimate and doubly warped product
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In this article, the universal covering of a manifold with positive constant curvatures was shown to be O(mathbf R^2\times (-c,c)$ with a doubly warped product metric.Abstract:
In this paper, we give an affirmative answer to Gromov's conjecture ([3, Conjecture E]) by establishing an optimal Lipschitz lower bound for a class of smooth functions on orientable open $3$-manifolds with uniformly positive sectional curvatures. For rigidity we show that the universal covering of the given manifold must be $\mathbf R^2\times (-c,c)$ with some doubly warped product metric if the optimal bound is attained. This gives a characterization for doubly warped product metrics with positive constant curvature. As a corollary, we also obtain a focal radius estimate for immersed toruses in $3$-spheres with positive sectional curvatures.read more
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One-sided complete stable minimal surfaces
TL;DR: In this paper, it was shown that there are no complete one-sided stable minimal surfaces in the Euclidean 3-space, and that the genus of compact two-sided index one minimal surfaces can be found in non-negatively curved ambient spaces.
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Metric Inequalities with Scalar Curvature
Misha Gromov,Misha Gromov +1 more
TL;DR: In this article, the authors established several inequalities for manifolds with positive scalar curvature and, more generally, for the scalars curvature bounded from below, based on the Schoen-Yau descent method via minimal hypersurfaces.
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Rigidity of area-minimizing two-spheres in three-manifolds
TL;DR: In this paper, the authors considered a three-manifold with positive scalar curvature and showed that any area-minimizing surface in M is homeomorphic to either S2 or RP.
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Four Lectures on Scalar Curvature
Misha Gromov,Misha Gromov +1 more
TL;DR: In this article, the authors overview main topics and ideas in spaces with their scalar curvatures bounded from below, and present a more detailed exposition of several known and some new geometric constraints on Riemannian spaces implied by the lower bounds on their curvatures.