The isoperimetric and Willmore problems

Sobolev and isoperimetric inequalities with monomial weights

By using optimal mass transport theory we prove a sharp isoperimetric inequality in $${\textsf {CD}} (0,N)$$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. In the case of n-dimensional Riemannian manifolds with nonnegative Ricci curvature, we outline an alternative proof of the rigidity result of Brendle (Comm Pure Appl Math 2021:13717, 2021). As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate symmetrization techniques and optimal volume non-collapsing properties. The equality cases in the latter inequalities are also characterized by stating that sufficiently smooth, nonzero extremal functions exist if and only if the Riemannian manifold is isometric to the Euclidean space.

The classical isoperimetric inequality in Euclidean space. Three different approaches.- The curve shortening flow and isoperimetric inequalities on surfaces.- $H^k$-flows and isoperimetric inequalities.- Estimates on the Willmore functional and isoperimetric inequalities.- Singularities in the volume-preserving mean curvature flow.- Bounds on the Heegaard genus of a hyperbolic manifold.- The isoperimetric profile for small volumes.- Local existence of flows driven by the second fundamental form and formation of singularities.- Invariance properties.- Singular behaviour of convex surfaces.- Convexity estimates.- Rescaling near a singularity.- Cylindrical and gradient estimates.- Mean curvature flow with surgeries.

This is a continuation of our previous work [Preprint, 2008, http://arxiv.org/abs/0712.4092]. It is well known that various isoperimetric inequalities imply their functional ‘counterparts’, but in general this is not an equivalence. We show that under certain convexity assumptions (for example, for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz–Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz'ya and extended by Barthe–Cattiaux–Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. As another application, we show that under our convexity assumptions, q-log-Sobolev inequalities (q ∈ [1, 2]) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry–Ledoux and Bobkov–Zegarliński. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvature–dimension condition of Bakry–Émery.

Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities

Isoperimetric inequalities on curved surfaces

Analysis and Algebra on Differentiable Manifolds

In this chapter we will determine cup products in the cohomology of the real, complex, and quaternionic projective spaces. The cup products (mod 2) in real projective spaces will be used to prove the famous Borsuk—Ulam theorem. Then we will introduce the mapping cone of a continuous map, and use it to define the Hopf invariant of a map f : S 2n-1 → S n. The proof of existence of maps of Hopf invariant 1 will depend on our determination of cup products in the complex and quaternionic projective plane.KeywordsProjective SpaceCommutative DiagramQuotient SpaceMapping ConeMapping CylinderThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We discuss an approach, based on the Brunn–Minkowski inequality, to isoperimetric and analytic inequalities for probability measures on Euclidean space with logarithmically concave densities. In particular, we show that such measures have positive isoperimetric constants in the sense of Cheeger and thus always share Poincaré-type inequalities. We then describe those log-concave measures which satisfy isoperimetric inequalities of Gaussian type. The results are precised in dimension 1.

Uniform tail-decay of Lipschitz functions implies Cheeger's isoperimetric inequality under convexity assumptions

A 2006 conjecture of Antunes and Freitas is addressed connecting the scaling-invariant polygonal isoperimetric and principal frequency deficits for triangles. This yields a quantitative polygonal Faber–Krahn inequality for triangles with an explicit constant. Furthermore, a problem mentioned in the 1951 book “Isoperimetric Inequalities In Mathematical Physics” by Pólya and Szegö is addressed: a formula is given for the principal frequency of a triangle. Moreover, a space of polygons is constructed for the classical Pólya and Szegö problem: in 1947, Pólya proved that if n = 3, 4 the regular polygon Pn minimizes the principal frequency of an n-gon with given area α > 0 and suggested that the same holds when n ≥ 5. In 1951, Pólya and Szegö discussed the possibility of counterexamples. This paper constructs explicit (2n − 4)–dimensional polygonal manifolds M(n,α) and proves the existence of a computable N ≥ 5 such that for all n ≥ N, the admissible n-gons are given via M(n,α)and there exists an explicit set An(α)⊂M(n,α) such that Pn has the smallest principal frequency among n-gons in An(α). Inter-alia when n ≥ 3, a formula is proved for the principal frequency of a convex P∈M(n,α)in terms of an equilateral n-gon with the same area; and, the set of equilateral polygons is proved to be an (n − 3)–dimensional submanifold of the (2n − 4)–dimensional manifold M(n,α)near Pn. The techniques involve a partial symmetrization, tensor calculus, the spectral theory of circulant matrices, and W2,p/BMO estimates. Last, an application is given in the context of electron bubbles.

On the isoperimetric problem with respect to a mixed Euclidean–Gaussian density

On the isoperimetric problem with double density

Higher-order Sobolev embeddings and isoperimetric inequalities

Let Σ be a k-dimensional complete proper minimal submanifold in the Poincaré ball model Bn of hyperbolic geometry. If we consider Σ as a subset of the unit ball Bn in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold Σ and the ideal boundary ∂ ∞ Σ $\partial _\infty \Sigma $ , say Vol ℝ ( Σ ) $\operatorname{Vol}_{\mathbb {R}}(\Sigma )$ and Vol ℝ ( ∂ ∞ Σ ) $\operatorname{Vol}_{\mathbb {R}}(\partial _\infty \Sigma )$ , respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if Vol ℝ ( ∂ ∞ Σ ) ≥ Vol ℝ ( 𝕊 k - 1 ) $\operatorname{Vol}_{\mathbb {R}}(\partial _\infty \Sigma ) \ge \operatorname{Vol}_{\mathbb {R}}(\mathbb {S}^{k-1})$ , then Σ satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such Σ, we further obtain a sharp lower bound for the Euclidean volume Vol ℝ ( Σ ) $\operatorname{Vol}_{\mathbb {R}}(\Sigma )$ , which is an extension of Fraser–Schoen and Brendle's recent results to hyperbolic space. Moreover we introduce the Möbius volume of Σ in Bn to prove an isoperimetric inequality via the Möbius volume for Σ.