Search

The focus of this paper is on a volume form defined on a pseudoconvex hypersurface M in a complex Calabi–Yau manifold (that is, a complex n -manifold with a nowhere-vanishing holomorphic n -form). We begin by defining this volume form and observing that it can be viewed as a generalization of the affine-invariant volume form on a convex hypersurface in R n . We compute the first variation, which leads to a similar generalization of the affine mean curvature. In Sec. 2, we investigate the constrained variational problem, for pseudoconvex hypersurfaces M bounding compact domains Ω ⊂ Z. That is, we study critical points of the volume functional A (M) where the ordinary volume V (Ω) is fixed. The critical points are analogous to constant mean curvature submanifolds. We find that Sasaki–Einstein hypersurfaces satisfy the condition, and in particular the standard sphere S 2 n − 1 ⊂ C n does. The main work in the paper comes in Sec. 3 where we compute the second variation about the sphere. We find that it is negative in "most" directions but non-negative in directions corresponding to deformations of S 2 n − 1 by holomorphic diffeomorphisms. We are led to conjecture a "minimax" characterization of the sphere. We also discuss connections with the affine geometry case and with Kähler–Einstein geometry. Our original motivation for investigating these matters came from the case n = 3 and the embedding problem studied in our previous paper [S. Donaldson and F. Lehmann, Closed 3-forms in five dimensions and embedding problems, preprint (2022), arXiv:2210.16208]. There are some special features in this case. The volume functional can be defined without reference to the embedding in Z using only a closed "pseudoconvex" real 3 -form on M. In Sec. 4, we review this and develop some of the theory from the point of the symplectic structure on exact 3 -forms on M and the moment map for the action of the diffeomorphisms of M. [ABSTRACT FROM AUTHOR]

In this paper a new generalized norm is defined and Riesz type inequalities for harmonic functions on the unit disk are discussed by applying it. Also sharp constants are obtained for certain special values considered in the reverse case of the standard Riesz inequality. [ABSTRACT FROM AUTHOR]

In this paper, we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centred at the origin is the only minimizer of such a functional for certain values of the mass. We prove that this is the case in dimension 2 while in higher dimension the situation is different. In fact, for small values of mass, the ball centred at the origin is a local minimizer, while for larger values the ball is a maximizer among convex sets with a uniform bound on the curvature. [ABSTRACT FROM AUTHOR]

We consider Steklov eigenvalues of nearly hyperspherical domains in Rd+1 with d≥3. In previous work, treating such domains as perturbations of the ball, we proved that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term of the asymptotic expansion and show that the first-order perturbations are eigenvalues of a Hermitian matrix, whose entries can be written explicitly in terms of Pochhammer's and Wigner 3j -symbols. We analyse the asymptotic expansion and show the following isoperimetric results among domains with fixed volume: (i) for an infinite subset of Steklov eigenvalues, the ball is not optimal and (ii) for a different infinite subset of Steklov eigenvalues, the ball is a stationary point. [ABSTRACT FROM AUTHOR]

In this paper we will establish a new Pólya–Szegö type inequality for a weighted gradient of a function on R 2 with respect to a weighted area. In order to do that we need to study an isoperimetric problem for the weighted area. We then apply the inequality to prove embedding theorems for weighted Sobolev spaces and to calculate the best constant in the Sobolev imbedding theorems. In our upcoming manuscript the obtained results in this note will be used to study boundary value problems for semilinear degenerate elliptic equations, see Luyen et al. (). [ABSTRACT FROM AUTHOR]

Large scale properties of Riemannian manifolds, in particular, those properties preserved by quasi-isometries, can be studied using discrete structures which approximate the manifolds. In a sequence of papers, M. Kanai proved that, under mild conditions, many properties are preserved by a certain (quasi-isometric) graph approximation of a manifold. One of these properties is p-parabolicity. A manifold M (respectively, a graph G) is said to be p-parabolic if all positive p-superharmonic functions on M (resp. G) are constant. This is equivalent to not having p-Green's function (i.e. a positive fundamental solution of the p-Laplace-Beltrami operator). Herein we study directly the p-parabolicity on graphs. We obtain some characterizations in terms of graph decompositions. Also, we give necessary and sufficient conditions for a uniform hyperbolic graph to be p-parabolic in terms of its boundary at infinity. Finally, we prove that if a uniform hyperbolic graph satisfies the (Cheeger) isoperimetric inequality, then it is non-p-parabolic for every 1 < p < ∞ . [ABSTRACT FROM AUTHOR]

We prove Michael-Simon type Sobolev inequalities for n-dimensional submanifolds in (n + m) -dimensional Riemannian manifolds with nonnegative kth intermediate Ricci curvature by using the Alexandrov-Bakelman-Pucci method. Here k = min (n - 1 , m - 1) . These inequalities extend Brendle’s Michael-Simon type Sobolev inequalities on Riemannian manifolds with nonnegative sectional curvature Brendle (Commun. Pure Appl. Math. 76(9), 2192–2218 (2022)) and Dong-Lin-Lu’s Michael-Simon type Sobolev inequalities on Riemannian manifolds with asymptotically nonnegative sectional curvature Dong et al. (Sobolev inequalities in manifolds with asymptotically nonnegative curvature, 2022) to the k-Ricci curvature setting. In particular, a simple application of these inequalities gives rise to some isoperimetric inequalities for minimal submanifolds in Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We investigate heat kernel estimates of the form pt(x,x) ≥ cxt−α, for large enough t, where α and cx are positive reals and cx may depend on x, on manifolds having at least one end with a polynomial volume growth. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if u is a solution to the Liouville equation 0.1 Δ u + e 2 u = 0 in R 2 , then the diameter of R 2 under the conformal metric g = e 2 u δ is bounded below by π . Here δ is the Euclidean metric in R 2 . Moreover, we explicitly construct a family of solutions to (0.1) such that the corresponding diameters of R 2 range over [ π , 2 π) . We also discuss supersolutions to (0.1). We show that if u is a supersolution to (0.1) and ∫ R 2 e 2 u d x < ∞ , then the diameter of R 2 under the metric e 2 u δ is less than or equal to 2 π . For radial supersolutions to (0.1), we use both analytical and geometric approaches to prove some inequalities involving conformal lengths and areas of disks in R 2 . We also discuss the connection of the above results with the sphere covering inequality in the case of Gaussian curvature bounded below by 1. Higher dimensional generalizations are also discussed. [ABSTRACT FROM AUTHOR]

We define, for any graph G = (V , E) , a boundary ∂ G ⊆ V. The definition coincides with what one would expected for the discretization of (sufficiently nice) Euclidean domains and contains all vertices from the Chartrand, Erwin, Johns & Zhang boundary. Moreover, it satisfies an isoperimetric principle stating that graphs with many vertices have a large boundary unless they contain long paths: we show that for graphs with maximaldegree Δ | ∂ G | ≥ 1 2 Δ | V | diam (G) . For graphs discretizing Euclidean domains, one has diam (G) ∼ | V | 1 / d and recovers the scaling of the classical Euclidean isoperimetric principle. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two-sided estimates for the Euclidean moment of inertia I2(G) of a convex domain G on the plane in terms of geometric characteristics of this domain similar to the Pólya–Szegő and Makai inequalities for the torsional rigidity. [ABSTRACT FROM AUTHOR]

The affine quermassintegrals associated to a convex body in \mathbb {R}^n are affine-invariant analogues of the classical intrinsic volumes from the Brunn–Minkowski theory, and thus constitute a central pillar of Affine Convex Geometry. They were introduced in the 1980's by E. Lutwak, who conjectured that among all convex bodies of a given volume, the k-th affine quermassintegral is minimized precisely on the family of ellipsoids. The known cases k=1 and k=n-1 correspond to the classical Blaschke–Santaló and Petty projection inequalities, respectively. In this work we confirm Lutwak's conjecture, including characterization of the equality cases, for all values of k=1,\ldots,n-1, in a single unified framework. In fact, it turns out that ellipsoids are the only local minimizers with respect to the Hausdorff topology. For the proof, we introduce a number of new ingredients, including a novel construction of the Projection Rolodex of a convex body. In particular, from this new view point, Petty's inequality is interpreted as an integrated form of a generalized Blaschke–Santaló inequality for a new family of polar bodies encoded by the Projection Rolodex. We extend these results to more general L^p-moment quermassintegrals, and interpret the case p=0 as a sharp averaged Loomis–Whitney isoperimetric inequality. [ABSTRACT FROM AUTHOR]

For a finitely generated group G and collection of subgroups P, we prove that the relative Dehn function of a pair (G, P) is invariant under quasi-isometry of pairs. Along the way, we show quasi-isometries of pairs preserve almost malnormality of the collection and fineness of the associated coned-off Cayley graphs. We also prove that for a cocompact simply connected combinatorial G-2-complex X with finite edge stabilisers, the combinatorial Dehn function is well defined if and only if the 1-skeleton of X is fine. We also show that if H is a hyperbolically embedded subgroup of a finitely presented group G, then the relative Dehn function of the pair (G, H) is well defined. In the appendix, it is shown that the Baumslag-Solitar group BS(k, l) has a well-defined Dehn function with respect to the cyclic subgroup generated by the stable letter if and only if neither k divides l nor l divides k. [ABSTRACT FROM AUTHOR]

We consider solutions to two boundary values problems for the Poisson equation on plane domains. We prove several estimates for integrals of solutions using geometric characteristics of domains. [ABSTRACT FROM AUTHOR]

In this article, we introduce a notion of capillary Schwarz symmetrization in the half-space. It can be viewed as the counterpart of the classical Schwarz symmetrization in the framework of capillary problem in the half-space. A key ingredient is a special anisotropic gauge, which enables us to transform the capillary symmetrization to the convex symmetrization introduced in Alvino et al. https:/doi.org/10.1016/S0294-1449(97)80147-3.

In this paper, we introduce the Laguerre bounded variation space and the Laguerre perimeter, thereby investigating their properties. Moreover, we prove the isoperimetric inequality and the Sobolev inequality in the Laguerre setting. As applications, we derive the mean curvature for the Laguerre perimeter.

We are interested in sharp functional inequalities for the coherent state transform related to the Wehrl conjecture and its generalizations. This conjecture was settled by Lieb in the case of the Heisenberg group, Lieb and Solovej for SU(2), and Kulikov for SU(1, 1) and the affine group. In this article, we give alternative proofs and characterize, for the first time, the optimizers in the general case. We also extend the recent Faber-Krahn-type inequality for Heisenberg coherent states, due to Nicola and Tilli, to the SU(2) and SU(1, 1) cases. Finally, we prove a family of reverse Hölder inequalities for polynomials, conjectured by Bodmann.

In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian manifolds with Ric ≥ (n − 1) κ , κ = 0 or 1 , which extends the results in [A. Alvino, C. Nitsch and C. Trombetti, A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions, Comm. Pure Appl. Math. 76(3) (2023) 585–603]. Furthermore, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel–Daners inequality for Robin Laplacian. [ABSTRACT FROM AUTHOR]

A Finsler analog of the Lobachevsky plane is the Lie group of proper affine transformations of the real line with a left-invariant Finsler structure generated by a convex compact set in the Lie algebra with the origin in its interior. We consider the isoperimetric problem on this Lie group, with the volume form also taken to be left-invariant. This problem is formulated as an optimal control problem. Applying the Pontryagin maximum principle, we find the optimal isoperimetric loops in an explicit form in terms of convex trigonometry functions. We also present a generalized isoperimetric inequality in a parametric form. [ABSTRACT FROM AUTHOR]

For a positive integer s, the projection body of an s-concave function f : ℝ n → [ 0 , + ∞) , a convex body in the (n + s)-dimensional Euclidean space ℝ n + s , is introduced. Associated inequalities for s-concave functions, such as, the functional isoperimetric inequality, the functional Petty projection inequality and the functional Loomis-Whitney inequality are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we prove a class of weighted isoperimetric inequalities for bounded domains in hyperbolic space by using the isoperimetric inequality with log-convex density in Euclidean space. As a consequence, we remove the horo-convex assumption of domains in a weighted isoperimetric inequality proved by Scheuer-Xia [Trans. Amer. Math. Soc. 372 (2019), pp. 6771–6803]. Furthermore, we prove weighted isoperimetric inequalities for star-shaped domains in warped product manifolds. Particularly, we obtain a weighted isoperimetric inequality for star-shaped hypersurfaces lying outside a certain radial coordinate slice in the anti-de Sitter-Schwarzschild manifold. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for any bounded set of finite perimeter \Omega \subset \mathbb {R}^n, we can choose smooth sets E_k \Subset \Omega such that E_k \rightarrow \Omega in L^1 and \begin{align*} \limsup _{i \rightarrow \infty } P(E_i) \le P(\Omega)+C_1(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*} In the above \Omega ^1 is the measure-theoretic interior of \Omega, P(\cdot) denotes the perimeter functional on sets, and C_1(n) is a dimensional constant. Conversely, we prove that for any sets E_k \Subset \Omega satisfying E_k \rightarrow \Omega in L^1, there exists a dimensional constant C_2(n) such that the following inequality holds: \begin{align*} \liminf _{k \rightarrow \infty } P(E_k) \ge P(\Omega)+ C_2(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*} In particular, these results imply that for a bounded set \Omega of finite perimeter, \begin{align*} \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1)=0 \end{align*} holds if and only if there exists a sequence of smooth sets E_k such that E_k \Subset \Omega, E_k \rightarrow \Omega in L^1 and P(E_k) \rightarrow P(\Omega). [ABSTRACT FROM AUTHOR]

We study the Riemannian quantitive isoperimetric inequality. We show that a direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the Łojasiewicz-Simon inequality. The main novelty of our work is that in all our results we do not require any a priori knowledge on the structure/shape of the minimizers. [ABSTRACT FROM AUTHOR]

In 1965, B. A. Troesch solved the isoperimetric sloshing problem of determining the container shape that maximizes the fundamental sloshing frequency among two classes of shallow containers: symmetric canals with a given free surface width and cross-sectional area, and radially symmetric containers with a given rim radius and volume (Commun Pure Appl Math 18(1–2):319–338, 1965, ). Here, we extend these results in two ways: (i) we consider surface tension effects on the fluid free surface, assuming a flat equilibrium free surface together with a pinned contact line, and (ii) we consider sinusoidal waves traveling along the canal with wavenumber α ≥ 0 and spatial period 2 π / α ; two-dimensional sloshing corresponds to the case α = 0 . Generalizing our recent variational characterization of fluid sloshing with surface tension to the case of a pinned contact line, we derive the pinned-edge linear shallow sloshing problem, which is an eigenvalue problem for a generalized Sturm-Liouville system. In the case without surface tension, we show that the optimal shallow canal is a rectangular canal for any α > 0 . In the presence of surface tension, we solve for the maximizing cross-section explicitly for shallow canals with any given α ≥ 0 and shallow radially symmetric containers with m azimuthal nodal lines, m = 0 , 1 . Our results reveal that the squared maximal sloshing frequency increases considerably as surface tension increases. Interestingly, both the optimal shallow canal for α = 0 and the optimal shallow radially symmetric container are not convex. As a consequence of our explicit solutions, we establish convergence of the maximizing cross-sections, as surface tension vanishes, to the maximizing cross-sections without surface tension. [ABSTRACT FROM AUTHOR]

In this article, we study scaling laws for simplified multi-well nucleation problems without gauge invariances which are motivated by models for shape-memory alloys. Seeking to explore the role of the order of lamination on the energy scaling for nucleation processes, we provide scaling laws for various model problems in two and three dimensions. In particular, we discuss (optimal) scaling results in the volume and the singular perturbation parameter for settings in which the surrounding parent phase is in the first-, the second- and the third-order lamination convex hull of the wells of the nucleating phase. Furthermore, we provide a corresponding result for the setting of an infinite order laminate which arises in the context of the Tartar square. In particular, our results provide isoperimetric estimates in situations in which strong nonlocal anisotropies are present. [ABSTRACT FROM AUTHOR]

In this article, we prove Talenti's comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber–Krahn inequality for the first eigenvalue of Dirichlet Laplacian, L 1 - and L ∞ -moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian. [ABSTRACT FROM AUTHOR]

It is well known that among all closed bounded curves in the plane with the given perimeter, the circle encloses the maximum area. There are many proofs in the literature. In this article, we have provided a new proof using some ideas of Demar [5]. [ABSTRACT FROM AUTHOR]

We introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a planar bounded simply-connected C 2 -smooth open set. The considered lower-order perturbation corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of a bounded convex planar set under the constraint on the maximum of the curvature of the boundary with the maximizer being the exterior of the disk. The isoperimetric inequality is proved under the assumption that to the lowest eigenvalue for the exterior of the disk corresponds a radial eigenfunction. We provide a sufficient condition in terms of the tension parameter and the radius for this property to hold. [ABSTRACT FROM AUTHOR]

Sharp upper and lower bounds for the isoperimetric deficit of triangles or parallelograms with the minimal annulus of radii R and r are given. [ABSTRACT FROM AUTHOR]

Given a bounded open set Ω ⊆ ℝn, we consider the eigenvalue problem for a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of Ω. We prove that the second eigenvalue λ2(Ω) is always strictly larger than the first eigenvalue λ1(B) of a ball B with volume half of that of Ω. This bound is proven to be sharp, by comparing to the limit case in which Ω consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity. [ABSTRACT FROM AUTHOR]

We develop a new technique for proving concentration inequalities which relate the variance and influences of Boolean functions. Using this technique, we Settle a conjecture of Talagrand (Combinatorica 17(2):275–285, 1997), proving that ∫ - 1 , 1 n h f x d μ x ≥ C · Var f · log 1 ∑ Inf i 2 f 1 / 2 , where h f x is the number of edges at x along which f changes its value, μ x is the uniform measure on - 1 , 1 n , and Inf i f is the influence of the i-th coordinate. Strengthen several classical inequalities concerning the influences of a Boolean function, showing that near-maximizers must have large vertex boundaries. An inequality due to Talagrand states that for a Boolean function f, Var f ≤ C ∑ i = 1 n Inf i f 1 + log 1 / Inf i f . We give a lower bound for the size of the vertex boundary of functions saturating this inequality. As a corollary, we show that for sets that satisfy the edge-isoperimetric inequality or the Kahn–Kalai–Linial inequality up to a constant, a constant proportion of the mass is in the inner vertex boundary. Improve a quantitative relation between influences and noise stability given by Keller and Kindler. Our proofs rely on techniques based on stochastic calculus, and bypass the use of hypercontractivity common to previous proofs. [ABSTRACT FROM AUTHOR]

Using Minkowski symmetrization, we obtain some sharp isoperimetric inequalities involving the mean width and the volume of the projection body (or the polar body of projection body) of an origin-symmetric convex body in R n . [ABSTRACT FROM AUTHOR]

In this paper, we obtain the Faber-Krahn inequality for the first eigenvalue of the Robin Laplacian on bounded domain in Riemannian manifolds whose Ricci curvature satisfies Ric g ≥ (n − 1). The Faber-Krahn inequality also holds for the Robin Laplacian on bounded domain in hyperbolic space H n. [ABSTRACT FROM AUTHOR]

In this short exposition, we provide a simplified proof of Buser’s result for Cheeger’s isoperimetric constant. We also provide a comprehensive approach on how to obtain volume estimates for smooth hypersurfaces. [ABSTRACT FROM AUTHOR]

We establish generalizations of the classical Polya–Szegö and Makai inequalities which estimate the torsional rigidity of a convex domain. The main idea of the proof is to apply a new exact isoperimetric inequality for Euclidean moments of a domain with respect to its boundary. This inequality has a wide class of extremal domains and is of independent interest itself. [ABSTRACT FROM AUTHOR]

We consider the Saint Venant functional for the torsional rigidity in simply connected plane domains. We prove new lower and upper estimates of using integrals of the conformal radius defined at any point of the domain and considered as a function. Applications of Davenport's formula for the torsional rigidity are discussed. In addition, we present a short https://doi.org/ of the Saint Venant–Pólya isoperimetric inequality. [ABSTRACT FROM AUTHOR]

Motivated by the celebrated work of Lutwak, Yang and Zhang [1] on (p,q)\left(p,q)-mixed volumes and that of Feng and He [2] on (p,q)\left(p,q)-mixed geominimal surface areas, we in the present paper establish and confirm the affine isoperimetric and Brunn-Minkowski-type inequalities with respect to (p,q)\left(p,q)-mixed geominimal surface areas.

In this paper, we consider a dilation type inequality on a weighted Riemannian manifold, which is classically known as Borell’s lemma in high-dimensional convex geometry. We investigate the dilation type inequality as an isoperimetric type inequality by introducing the dilation profile and estimate it by the one for the corresponding model space under lower weighted Ricci curvature bounds. We also explore functional inequalities derived from the comparison of the dilation profiles under the nonnegative weighted Ricci curvature. In particular, we show several functional inequalities related to various entropies.

Our main result gives an improved bound on the filling areas of curves in Banach spaces which are not closed geodesics. As applications we show rigidity of Pu's classical systolic inequality and investigate the isoperimetric constants of normed spaces. The latter has further applications concerning the regularity of minimal surfaces in Finsler manifolds. [ABSTRACT FROM AUTHOR]

We consider the magnetic Robin Laplacian with a negative boundary parameter on a bounded, planar C 2 -smooth domain. The respective magnetic field is homogeneous. Among a certain class of domains, we prove that the disk maximises the ground-state energy under the fixed perimeter constraint provided that the magnetic field is of moderate strength. This class of domains includes, in particular, all domains that are contained upon translations in the disk of the same perimeter and all centrally symmetric domains. [ABSTRACT FROM AUTHOR]

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$ -groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$ , the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$ , which implies hyperbolicity. [ABSTRACT FROM AUTHOR]

This note concerns Loomis–Whitney inequalities in Heisenberg groups H n : | K | ≲ ∏ j = 1 2 n | π j (K) | n + 1 n (2 n + 1) , K ⊂ H n. Here π j , j = 1 , ... , 2 n , are the vertical Heisenberg projections to the hyperplanes { x j = 0 } , respectively, and | · | refers to a natural Haar measure on either H n , or one of the hyperplanes. The Loomis–Whitney inequality in the first Heisenberg group H 1 is a direct consequence of known L p improving properties of the standard Radon transform in R 2 . In this note, we show how the Loomis–Whitney inequalities in higher dimensional Heisenberg groups can be deduced by an elementary inductive argument from the inequality in H 1 . The same approach, combined with multilinear interpolation, also yields the following strong type bound: ∫ H n ∏ j = 1 2 n f j (π j (p)) d p ≲ ∏ j = 1 2 n ‖ f j ‖ n (2 n + 1) n + 1 for all nonnegative measurable functions f 1 , ... , f 2 n on R 2 n . These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis–Whitney inequalities in H n , we mention the following sharper version of the classical geometric Sobolev inequality in H n : ‖ u ‖ 2 n + 2 2 n + 1 ≲ ∏ j = 1 2 n ‖ X j u ‖ 1 2 n , u ∈ B V (H n) , where X j , j = 1 , ... , 2 n , are the standard horizontal vector fields in H n . Finally, we also establish an extension of the Loomis–Whitney inequality in H n , where the Heisenberg vertical coordinate projections π 1 , ... , π 2 n are replaced by more general families of mappings that allow us to apply the same inductive approach based on the L 3 / 2 - L 3 boundedness of an operator in the plane. [ABSTRACT FROM AUTHOR]

We consider Steklov eigenvalues of three-dimensional, nearly spherical domains. In previous work, we have shown that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term of the asymptotic expansion, which can explicitly be written in terms of the Wigner 3-j symbols. We analyze the asymptotic expansion and prove the isoperimetric result that, if\ell is a square integer, the volume-normalized\ell th Steklov eigenvalue is stationary for a ball. [ABSTRACT FROM AUTHOR]