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Register a new userQuantitative stability for hypersurfaces with almost constant curvature in space forms
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The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures $f(k_1,ldots,k_{n-1})$ satisfying suitable conditions. In this paper we give sharp quantitative estimates of proximity to a single sphere for Alexandrov Soap Bubble Theorem in space forms when the curvature operator $f$ is close to a constant. Under an assumption that prevents bubbling, the proximity to a single sphere is quantified in terms of the oscillation of the curvature function $f$. Our approach provides a unified picture of quantitative studies of the method of moving planes in space forms.
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Alexandrovs soap bubble theorem asserts that spheres are the only connected closed embedded hypersurfaces in the Euclidean space with constant mean curvature. The theorem can be extended to space forms and it holds for more general functions of the principal curvatures. In this short review, we discuss quantitative stability results regarding Alexandrovs theorem which have been obtained by the author in recent years. In particular, we consider hypersurfaces having mean curvature close to a constant and we quantitatively describe the proximity to a single sphere or to a collection of tangent spheres in terms of the oscillation of the mean curvature. Moreover, we also consider the problem in a non local setting, and we show that the non local effect gives a stronger rigidity to the problem and prevents the appearance of bubbling.
We prove a converse to well-known results by E. Cartan and J. D. Moore. Let $fcolon M^n_ctoQ^{n+p}_{tilde c}$ be an isometric immersion of a Riemannian manifold with constant sectional curvature $c$ into a space form of curvature $tilde c$, and free of weak-umbilic points if $c>tilde{c}$. We show that the substantial codimension of $f$ is $p=n-1$ if, as shown by Cartan and Moore, the first normal bundle possesses the lowest possible rank $n-1$. These submanifolds are of a class that has been extensively studied due to their many properties. For instance, they are holonomic and admit B{a}cklund and Ribaucour transformations.
We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetricinequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the $n=2$ case of a conjecture of Gir~ao-Pinheiro.
We give a new proof of the generalized Minkowski identities relating the higher degree mean curvatures of orientable closed hypersurfaces immersed in a given constant sectional curvature manifold. Our methods rely on a fundamental differential system of Riemannian geometry introduced by the author. We develop the notion of position vector field, which lies at the core of the Minkowski identities.
We study the uniqueness of horospheres and equidistant spheres in hyperbolic space under different conditions. First we generalize the Bernstein theorem by Do Carmo and Lawson to the embedded hypersurfaces with constant higher order mean curvature. Then we prove two Bernstein type results for immersed hypersurfaces under different assumptions. Last, we show the rigidity of horospheres and equidistant spheres in terms of their higher order mean curvatures.
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