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Quantitative C 1 -stability of spheres in rank one symmetric spaces of non-compact type
Abstract
We prove that in any rank one symmetric space of non-compact type M ∈ { ℝ H n , ℂ H m , ℍ H m , 𝕆 H 2 } {M\in\{\mathbb{R}H^{n},\mathbb{C}H^{m},\mathbb{H}H^{m},\mathbb{O}H^{2}\}} , geodesic spheres are uniformly quantitatively stable with respect to small C 1 {C^{1}} -volume preserving perturbations. We quantify the gain of perimeter in terms of the W 1 , 2 {W^{1,2}} -norm of the perturbation, taking advantage of the explicit spectral gap of the Laplacian on geodesic spheres in M . As a consequence, we give a quantitative proof that for small volumes, geodesic spheres are isoperimetric regions among all sets of finite perimeter.
... This question, rooted in the seminal work of Fuglede [30], has a rich history in the study of isoperimetric problems. Rigorous results addressing similar questions have been established in various contexts, including the Euclidean setting [18,19,31,32,34,37], Riemannian manifolds [8,9,13,16,24,62], anisotropic energies [23,27,55,56], and weighted settings [17,20,33]. ...
We study the isoperimetric problem with a potential energy g in weighted by a radial density f and analyze the geometric properties of minimizers. Notably, we construct two counterexamples demonstrating that, in contrast to the classical isoperimetric case , the condition does not generally guarantee the global optimality of centered spheres. However, we demonstrate that centered spheres are globally optimal when both f and g are monotone. Additionally, we strengthen this result by deriving a sharp quantitative stability inequality.
This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on N-dimensional RCD(K,N) spaces (X,d,HN). Moreover, we obtain almost regularity theorems formulated in terms of the isoperimetric profile and enhanced consequences at the level of several functional inequalities. Most of our statements seem to be new even in the classical setting of smooth, non compact manifolds with lower Ricci curvature bounds. The synthetic theory plays a key role via compactness and stability arguments.
We provide an isoperimetric comparison theorem for small volumes in an n-dimensional Riemannian manifold with strong bounded geometry, as in Definition 2.3, involving the scalar curvature function. Namely in strong bounded geometry, if the supremum of scalar curvature function for some , then for small volumes the isoperimetric profile of is less then or equal to the isoperimetric profile of the complete simply connected space form of constant sectional curvature . This work generalizes Theorem 2 of [Dru02b] in which the same result was proved in the case where is assumed to be just compact. As a consequence of our result we give an asymptotic expansion in Puiseux's series up to the second nontrivial term of the isoperimetric profile function for small volumes. Finally, as a corollary of our isoperimetric comparison result, it is shown, in the special case of manifolds with strong bounded geometry, and that for small volumes the Aubin-Cartan-Hadamard's Conjecture in any dimension n is true.
For convex bodies D in R" the deviation d from spherical shape is estimated from above in terms of the (dimensionless) isoperimetric deficiency Δ of D as follows: d ≤ fΔ (for A sufficiently small). Here f is an explicit elementary function vanishing continuously at 0. The estimate is sharp as regards the order of magnitude of f. The dimensions n = 2 and 3 present anomalies as to the form of f. In the planar case n = 2 the result is contained in an inequality due to T. Bonnesen. A qualitative consequence of the present result is that there is stability in the classical isoperimetric problem for convex bodies D in Rn in the sense that, as D varies, d → 0 for Δ → 0. The proof of the estimate d ≤ f(Δ) is based on a related estimate in the case of domains (not necessarily convex) that are supposed a priori to be nearly spherical in a certain sense.
We generalize a result from Barbosa, doCarmo, and Eschenburg.
Hypersurfaces with constant mean curvature in a Riemannian manifold display many similarities with minimal hypersurfaces of They are both solutions to the variational problem of minimizing the area function for certain variations. In the first case, however, the admissible variations are only those that leave a certain volume function fixed (for precise definitions, see Sect. 2). This isoperimetric character of the variational problem associated to hypersurfaces of constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces.
We give a refinement of the quantitative isoperimetric inequality. We prove
that the isoperimetric gap controls not only the Fraenkel asymmetry but also
the oscillation of the boundary.
We introduce a new variational method for the study of stability in the isoperimetric inequality. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in . Second we positively answer to a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in in the small asymmetry regime. Comment: 19 pages
A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall.
This book deals with a class of mathematical problems which involve the minimization of the sum of a volume and a surface energy and have lately been refered to as 'free discontinuity problems'. The aim of this book is twofold: The first three chapters present all the basic prerequisites for the treatment of free discontinuity and other variational problems in a systematic, general, and self-contained way. In the later chapters, the reader is introduced to the theory of free discontinuity problems, to the space of special functions of bounded variation, and is presented with a detailed analysis of the Mumford-Shah image segmentation problem. Existence, regularity and qualitative properties of solutions are explained and a survey is given on the current knowledge of this challenging mathematical problem. Free discontinuity problems reveal a wide range of applications. The theory embodies classical problems, e.g. related to phase transitions, or fracture and plasticity in continuum mechanics, as well as more recent ones like edge detection in image analysis. This book provides the reader with a solid introduction to the field, written by principle contributors to the theory. The first half of the book contains a comprehensive and updated treatment of the theory of Functions of Bounded Variation and of the mathematical prerequisites of that theory, that is Abstract Measure Theory and Geometric Measure Theory.
In this paper, we investigate the regularity theory of codimension- 1 integer rectifiable currents that (almost)-minimize parametric elliptic functionals. While in the non-parametric case it follows by De Giorgi-Nash's Theorem that C¹,1 regularity of the integrand is enough to prove C¹,α regularity of minimizers, the present regularity theory for parametric functionals assume the integrand to be at least of class C2. In this paper, we fill this gap by proving that C1,1 regularity is enough to show that flat almost-minimizing currents are C1,α. As a corollary, we also show that the singular set has codimension greater than 2. Besides the result "per se", of particular interest we believe to be the approach used here: instead of showing that the standard excess function decays geometrically around every point, we construct a new excess with respect to graphs minimizing the nonparametric functional, and we prove that if this excess is sufficiently small at some radius R then it is identically zero at scale R/2. This implies that our current coincides with a minimizing graph there, hence it is of class C1,α.
The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. The theory of sets of finite perimeter provides a simple and effective framework. Topics covered include existence, regularity, analysis of singularities, characterization and symmetry results for minimizers in geometric variational problems, starting from the basics about Hausdorff measures in Euclidean spaces and ending with complete proofs of the regularity of area-minimizing hypersurfaces up to singular sets of codimension 8. Explanatory pictures, detailed proofs, exercises and remarks providing heuristic motivation and summarizing difficult arguments make this graduate-level textbook suitable for self-study and also a useful reference for researchers. Readers require only undergraduate analysis and basic measure theory.
A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate
for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially
Gromov’s proof of the isoperimetric inequality and the Brenier-McCann Theorem. A sharp quantitative version of the Brunn-Minkowski
inequality for convex sets is proved as a corollary.
We define a new class of submanifolds called pseudo-bubbles, defined by an equation weaker than constancy of mean curvature.
We show that in a neighborhood of each point of a Riemannian manifold, there is a unique family of concentric pseudo-bubbles
which contains all the pseudo-bubbles C
2,α
-close to small spheres. This permit us to reduce the isoperimetric problem for small volumes to a variational problem in
finite dimension.
We prove existence of regions minimizing perimeter under a volume constraint
in contact sub-Riemannian manifolds such that their quotient by the group of
contact transformations preserving the sub-Riemannian metric is compact.
. We give a new proof of the small excess regularity theorems for integer multiplicity rectifiable currents of arbitrary dimension and codimension minimizing an elliptic parametric variational integral. This proof does not use indirect blow-up arguments, it covers interior and boundary regularity, it applies to almost minimizing currents, and it gives an explicit and often optimal modulus of continuity for the derivative, i.e. for the tangent plane field of the almost minimizing currents. 0 Introduction The regularity theory for integer multiplicity rectifiable currents minimizing a parametric elliptic variational integral was initiated in the pioneering work of F.J. Almgren [Alm1], where the regularity is proved at interior support points with small cylindrical excess. Almgren's results are presented in F. Federer's monograph [Fe, Chap. 5]. E. Bombieri has given a somewhat simpler proof which also applies to approximately minimizing currents. This is important in order to be abble to...
- E Hebey
- S Helgason
Beweis des Satzes, dass die Kugel kleinere Oberfläche besitzt, als jeder andere Körper gleichen Volumens
- H A Schwarz