Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space

Mon, 26 Oct 2026 00:00:00 GMT

Tianci Luo, Yong Wei, Ruijia Zhang, Rong Zhou — We study quermassintegral-preserving curvature flows in ℍⁿ⁺¹, n≥3, driven by the curvature quotient (Eₖ/E_ℓ)^(1/(k-ℓ)). For 1≤ℓ

The quermassintegral inequalities for k-convex spiral-star-shaped domains

Mon, 26 Oct 2026 00:00:00 GMT

Junfang Li — We extend the quermassintegral inequalities of our earlier work from star-shaped domains to a substantially broader class of domains in Euclidean space, which we call spiral-star-shaped. A smooth domain is spiral-star-shaped if its boundary satisfies ⟨ X+A X,ν⟩>0 for some fixed skew-symmetric linear map A, where X is the position vector and ν is the outward unit normal. The vector field X+A X is conformal Killing, with dilation component X and rotational component A X; its trajectories combine expansion and rotation, giving rise to the spiral terminology. When A=0, it reduces to ordinary star-shapedness, while nonzero A allows domains that are not star-shaped in the usual radial sense. For smooth k-convex spiral-star-shaped domains, we prove long-time existence and exponential convergence to a round sphere for a locally constrained inverse curvature flow. The positive spiral support function provides the geometric control needed to treat initial hypersurfaces that need not be star-shaped in the usual sense. Monotonicity of the quermassintegrals along the flow yields sharp quermassintegral inequalities in this enlarged class, with equality if and only if the domain is a ball.

Quermassintegral inequalities for capillary hypersurfaces in a horoball

Thu, 22 Oct 2026 00:00:00 GMT

Jinyu Guo, Liangjun Weng — In this paper, we study the quermassintegral inequalities for capillary hypersurfaces in a horoball in hyperbolic space. We construct a locally constrained nonlinear curvature flow that preserves the k-th quermassintegral while monotonically increasing the l-th quermassintegral for any (1≤ l

Modified support measures and Alexandrov-Fenchel type inequalities for nonsmooth h-convex bodies in hyperbolic space

Tue, 13 Oct 2026 00:00:00 GMT

Xinxiu Zhang — We establish the Alexandrov-Fenchel inequality and its modified version for arbitrary, possibly nonsmooth, horospherically convex bodies in hyperbolic space ℍⁿ⁺¹, n≥ 2, and prove that equality in each inequality holds if and only if the h-convex body is a geodesic ball. A key ingredient is a rigidity theorem characterizing geodesic balls via proportionality between modified support measures of adjacent orders. Its proof combines the C^(1,α)∩ W^(2,k+1) approximation of the outer parallel hypersurfaces by smooth strictly h-convex hypersurfaces with the estimates for the total variation norm of the signed measure associated with the proportionality relation along the shifted curvature flows independent of initial curvature bounds. For 0≤ k≤ n-1, we construct weak flows preserving the k-th modified quermassintegral Wₖ with arbitrary h-convex initial bodies as Hausdorff limits of the smooth flows. The rigidity theorem above characterizes the limit shape of the flow and gives a positive lower bound on the rate of decrease of Wₖ₊₁ for a short time whenever the initial h-convex body is not a geodesic ball, which characterizes the equality cases in the modified Alexandrov-Fenchel inequality. For the Alexandrov-Fenchel inequality, we establish the Hausdorff stability independent of curvature bounds described by the Alexandrov-Fenchel deficits for h-convex bodies, which characterizes the equality cases in the Alexandrov-Fenchel inequality.

Sharp dynamical isoperimetric principle for mean curvature flow

Wed, 07 Oct 2026 00:00:00 GMT

Beomjun Choi, Wenkui Du, Seung Chul Park, Junseo Youn — In this paper, we establish a sharp dynamical isoperimetric principle for weak mean curvature flow. We prove that, among weak mean curvature flows with fixed area of initial smooth closed hypersurfaces, the standard smoothly shrinking spherical mean curvature flow uniquely maximizes the extinction time. More precisely, for the level set flow Kₜ starting from the boundary of a smooth bounded domain Ω⊂ℝⁿ⁺¹, as well as for the integral Brakke flow \μₜ\ with initial Radon measure μ₀=𝓗ⁿllcorner∂Ω, we establish the corresponding optimal extinction time estimates \beginequation* T_{\rm ext}(Ω),\, T^B_{\rm ext}(Ω) \leq \frac12n \left(\fracP( Ω){|{\mathbbS^n}|}\right)^{\frac2n}, \endequation* where P(Ω) is the perimeter of Ω representing the area of ∂Ω, and the equality holds if and only if Ω is a round ball and the flow is the standard multiplicity-one smoothly self-shrinking round sphere. In particular, we obtain the sharp L^p-estimates for the arrival time function of a smooth bounded mean convex domain. In addition, we also establish the sharp isoperimetric inequality for the parabolic measure of the space-time track filling X of outward minimizing level set flow starting from the boundary of a smooth bounded domain Ω⊂ ℝⁿ⁺¹: \beginequation* \mathcal H_par^n+2(X) \leq \fracπ {2n(n+2)^2|{\mathbbS^n}|^{{\frac2n}}} P(Ω)^{\fracn+2n}, \endequation* where the equality holds if and only if Ω is a round ball and the flow is standard smoothly shrinking round sphere.

Eventual Monotonicity of the Willmore Energy under Mean Curvature Flow

Wed, 16 Sep 2026 00:00:00 GMT

Yuyue Chen, Shiying Wu — We establish the eventual monotonicity of the Willmore deficit ∫_ΣH² dμ−16π along the mean curvature flow of smooth closed embedded strictly convex surfaces in ℝ³, thereby complementing Huisken's exponential convergence with an eventual monotonicity property. As a consequence, the Hawking mass is eventually nondecreasing along the contracting mean curvature flow, in contrast with the usual inclusion monotonicity expected of the Hawking mass. We also discuss totally umbilical solutions in three-dimensional space forms, where the monotonicity directions of the Willmore energy and the Hawking mass depend on the sign of the ambient sectional curvature.

Alexandrov–Fenchel inequalities for convex domains in the sphere

Sat, 12 Sep 2026 00:00:00 GMT

Tianci Luo, Yong Wei, Rong Zhou — We prove the Alexandrov–Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov–Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.

Sharp p-Capacity Estimates via Quermassintegrals in Hyperbolic Space

Tue, 18 Aug 2026 00:00:00 GMT

Xiaoshang Jin, Yao Wan, Jie Xiao — This paper establishes sharp upper bounds for p-capacities Cap_(12m+1, an interpolating radius combines the 2m-th curvature-excess radius with the L^∞ curvature scale, thereby linking the finite-moment and supremum regimes. Equality in the sharp comparisons characterizes geodesic balls.

The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere

Tue, 11 Aug 2026 00:00:00 GMT

Sara Albert-Niclòs, Esther Cabezas-Rivas, Shujing Pan — We introduce fully nonlinear curvature flows of horo-convex hypersurfaces in the sphere that preserve an arbitrarily prescribed spherical quermassintegral. The non-local normalization is defined relative to a fixed ambient origin, and we prove that the evolution is governed by a single smooth fixed-origin equation for all time. For monotone, homogeneous curvature functions satisfying concavity and inverse concavity, we establish preservation of horo-convexity, uniform curvature pinching, a direct estimate for the non-local coefficient. A Tso-type argument then yields global curvature bounds and long-time existence. We further prove exponential decay of the traceless second fundamental form and exponential C^∞ convergence to the geodesic sphere centered at the fixed origin whose radius is determined by the preserved quermassintegral.

The Penrose inequality with charge for 2-convex initial data sets

Fri, 31 Jul 2026 00:00:00 GMT

Tuan Dolmen — We prove the Penrose inequality with charge under the 2-convexity condition recently introduced by Dong. More precisely, given a complete, connected and asymptotically flat Einstein-Maxwell initial data set (M,g,k; E,B) satisfying the charged dominant energy and the 2-convexity conditions, with divergence-free electromagnetic vector fields (E,B) and a connected outermost past apparent horizon Σ that satisfies |Σ| ≥ 4πq² - where q is the total charge - we show that the following inequality for the ADM mass m holds: m≥ √((|Σ|)/16π) + q² √(fracπ|Σ|), with equality if and only if k ≡ 0 and (M,g;E,B) is isometric to a canonical slice of sub-extremal Reissner-Nordström spacetime. Building on Dong's proof of the uncharged case, we use his P-inverse mean curvature flow and its weak formulation, which only depends on (g,P) and hence applies to the charged setting unchanged. The novelty of our work is the modification of the monotonicity formula to account for the additional charge term. For time-symmetric data (k ≡ 0), the flow reduces to the classical inverse mean curvature flow and our monotonicity formula to Jang's monotonicity of the charged Hawking mass, recovering the charged Riemannian Penrose inequality.

Inverse Hessian Curvature Flow in Minkowski Space I: Cocompact Hypersurfaces

Fri, 12 Jun 2026 00:00:00 GMT

Dake Li, Zhizhang Wang, Shiqi Yin — In this paper, we investigate the cocompact inverse σₖ curvature flow in Minkowski space. We prove the longtime existence and convergence of this flow. As a consequence, quermassintegral inequalities are established.

On the structure of complete G₂-solitons

Thu, 04 Jun 2026 00:00:00 GMT

Haozhao Li, Yuanqing Ma, Kai Zheng — In this work, we establish compactness and regularity results for complete gradient Laplacian solitons of closed G₂-structures. Under a lower scalar-curvature bound and a distance-dependent bound on the gradient of the soliton potential, we prove pointed measured Gromov-Hausdorff compactness. With a uniform lower bound for the localised Perelman entropy, the Gromov-Hausdorff convergence improves to pointed C^(1,α) convergence. Our principal result shows that this C^(1,α) convergence upgrades to smooth convergence on the regular set. More precisely, after passing to a subsequence, the metrics, the defining positive 3-forms, and the soliton potentials converge smoothly on every compact subset of the regular set, and the limiting data define a gradient Laplacian soliton. The proof develops a local entropy method adapted to G₂-solitons. Since the available C^(1,α) control does not directly close an elliptic bootstrap for the G₂-soliton, and no suitable pseudolocality theorem is available in this setting, we instead use the localised Perelman's functionals. These yield an entropy ε-regularity theorem and a gap theorem for scalar-flat solitons. On the regular set, pointed C^(1,α) convergence gives an almost-Euclidean local isoperimetric inequality, which in turn verifies the required small-entropy condition automatically. The resulting curvature bounds are then combined with G₂-specific differential identities and quantitative interior estimates to control the soliton data. Finally, at the critical exponent in dimension seven, we show that a uniform weighted L^(7/2) -curvature bound then yields pointed C^∞ compactness.

Sharp Gaussian Isoperimetry along a Ricci Flow

Wed, 20 May 2026 00:00:00 GMT

Robert Koirala — We prove the sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along a Ricci flow via a monotonicity formula. As consequences, we obtain the exact Gaussian enlargement theorem and a Gaussian-quantile two-set concentration estimate. In particular, this recovers the exponential concentration estimate of Hein–Naber from a sharper isoperimetric profile. We also derive Gaussian rearrangement inequalities, recover the sharp Hein–Naber log-Sobolev inequality, and identify the universal Gaussian-model constants in Bamler's L^p-Poincaré inequalities. Further applications include Gaussian-profile localization near Bamler's Hₙ-centers, convex-order and moment estimates for logarithmic derivatives of the conjugate heat kernel, reverse hypercontractivity, entropy-regular profile stability, and a path-space Bobkov inequality.

Capillary quermassintegral inequalities in the unit ball

Fri, 17 Apr 2026 00:00:00 GMT

Shujing Pan, Julian Scheuer — This paper is about hypersurfaces with boundary lying in the Euclidean unit ball, which meet the unit sphere at a fixed angle θ∈(0,fracπ2]. Such hypersurfaces are called θ-capillary hypersurfaces and for those we introduce a new notion of convexity, which we call θ-horocap-convexity. For such hypersurfaces, we prove the convergence of a curvature flow of Guan/Li type with capillary boundary. Remarkably, we prove this result for a class of curvature functions which include all quotients of symmetric polynomials and, as a consequence, we obtain the full set of quermassintegral inequalities in the θ-horocap-convex case. In the strictly horocap-convex setting, we employ the flow to prove the geometric inequalities, while for the horocap-convex case and the characterization of the equality case, we develop new arguments which are interesting in their own right.

Sharp Minkowski-Type Inequality in Cartan-Hadamard 3-Spaces

Fri, 20 Mar 2026 00:00:00 GMT

Fang Hong — In this paper, we prove a sharp Minkowski-type inequality in Cartan-Hadamard 3-spaces using harmonic mean curvature flow and improve the known estimates for total mean curvature in hyperbolic 3-space. In particular, we sharpen Ghomi-Spruck's result in by retaining the volume contribution in the monotonicity argument. As a corollary, we obtain a comparison theorem relating the total mean curvature of convex surfaces in Cartan-Hadamard 3-spaces to their enclosed volume.

Locally constrained inverse curvature flow and Alexandrov-Fenchel type inequalities in de Sitter space

Mon, 22 Dec 2025 00:00:00 GMT

Kuicheng Ma — In this paper, we study the behavior of some locally constrained inverse curvature flow in de Sitter space, with initial value any closed spacelike k-convex hypersurface satisfying some pinching condition. Assume further the Heintze-Karcher inequality for any closed spacelike mean convex hypersurface in de Sitter space, we derive a class of Alexandrov-Fenchel inequalities.

Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space

Sat, 01 Nov 2025 00:00:00 GMT

Yaoting Gui, Yuqiao Li, Jun Sun — In this paper, we investigate the stability of the volume preserving mean curvature flow (VPMCF) and area preserving mean curvature flow (APMCF) in the Schwarzschild space. We show that if the initial hypersurface is sufficiently close to a coordinate sphere, these flows exist globally and converge smoothly to a constant mean curvature (CMC) hypersurface, namely a coordinate sphere. For asymptotically Schwarzschild space, if the initial hypersurface has pinched curvature outside of some large compact set, or more orecisely sufficiently close to an isoperimetric hypersurface, outside of some large compact set in C^2 sense, we will apply similar method combined with the center manifold analysis to see that the flow still exists for all time and converges to CMC hypersurface exponentially fast. This in particular gives an existence result for a CMC hypersurface in asymptotically flat space.

The p-th dual Minkowski problem for the k-torsional rigidity corresponding to a k-Hessian equation

Wed, 29 Oct 2025 00:00:00 GMT

Xia Zhao, Peibiao Zhao — The study of the dual curvature measures [Y. Huang, E. Lutwak, D. Yang & G. Y. Zhang, Acta. Math. 216 (2016): 325-388], which connects the cone-volume measure and Aleksandrov's integral curvature, and has created a precedent for the theoretical research of the dual Brunn-Minkowski theory. Motivated by the foregoing groundbreaking works, the present paper introduces the p-th dual k-torsional rigidity associated with a k-Hessian equation and establishes its Hadamard variational formula with 1≤ k≤ n-1, which induces the p-th dual k-torsional measure. Further, based on the p-th dual k-torsional measure, this article, for the first time, proposes the p-th dual Minkowski problem of the k-torsional rigidity which can be equivalently converted to a nonlinear partial differential equation in smooth case: \beginalign f(x)=τ(|\nabla h|^2+h^2)^{\fracp-n2}h_Ω(x)|Du(ν^-1_Ω(x))|^k+1σ_n-k(h_ij(x)+h_Ω(x)δ_ij), \endalign where τ>0 is a constant, f is a positive smooth function defined on Sⁿ⁻¹ and σₙ₋ₖ is the (n-k)-th elementary symmetric function of the principal curvature radii. We confirm the existence of smooth non-even solution to the p-th dual Minkowski problem of the k-torsional rigidity for p

On the geometry and uniqueness of asymptotically locally hyperbolic static vacuum black holes

Mon, 22 Sep 2025 00:00:00 GMT

Brian Harvie, Ye-Kai Wang — We establish several geometric characterizations and rigidity results for 3-dimensional asymptotically locally hyperbolic (ALH) static spaces with horizon boundary. Notably, a 3-dimensional ALH static space with toroidal infinity and strictly non-spherical horizons is isometric to a toroidal Kottler metric. Furthermore, we show that the surface gravity of a static horizon with spherical infinity is bounded below by √3, with equality achieved only by the critical AdS-Schwarzschild metric. Consequently, Poincaré-Einstein fillings of S² × S¹(λ) arising from these spaces have length parameter λ≤ (√3)⁻¹, which supports a recent conjecture of Chang-Yang-Zhang[18]. Finally, static horizons with hyperbolic infinity and non-negative Chruściel-Herzlich mass obey the reverse Riemannian Penrose inequality. In conjunction with work of Ge-Wang-Wu-Xia [29], we use this fact to obtain uniqueness of static ALH graphs with hyperbolic infinities. These results follow from a generalization of the Minkowski inequality in AdS-Schwarzschild space due to Brendle-Hung-Wang [15]. Using optimal coefficients for the sub-static Heintze-Karcher inequality from [24], we construct a new monotone quantity under inverse mean curvature flow (IMCF) in static spaces with negative cosmological constant. Another fundamental tool developed in this paper is a regularity theorem for IMCF in ALH manifolds. Specifically, we prove that a weak solution of IMCF in an ALH 3-manifold with horizon boundary is eventually smooth. This extends the regularity theorem for a spherical infinity due to Shi-Zhu [52].

Inverse curvature flows for capillary hypersurfaces in the unit ball

Wed, 16 Jul 2025 00:00:00 GMT

Shujing Pan, Bo Yang — In this paper, we study inverse curvature flows for strictly convex, capillary hypersurfaces in the unit Euclidean ball. We establish the existence and convergence results for a class of such flows. As an application, we derive a family of Alexandrov Fenchel inequalities for weakly convex hypersurfaces with free boundary.

A locally constrained inverse Hessian quotient flow in de Sitter space

Tue, 15 Jul 2025 00:00:00 GMT

Kuicheng Ma — In this paper, an Alexandrov-Fenchel inequality is established for closed 2-convex spacelike hypersurface in de Sitter space by investigating the behavior of the locally constrained inverse curvature flow \beginalign \frac{\partial }{\partial t}x=\bigg(u-\fracλ'E_1E_2\bigg)ν,\nonumber \endalign which provides a partial answer to the conjecture raised by Hu and Li in.

New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

Sat, 28 Jun 2025 00:00:00 GMT

Jie Wu — In this paper, we establish a broad class of new sharp Alexandrov-Fenchel inequalities involving general convex weight functions for static convex hypersurfaces in hyperbolic space. Additionally, we derive new weighted Minkowski-type inequalities for static convex hypersurfaces in hyperbolic space ℍⁿ and for convex hypersurfaces in the sphere Sⁿ. The tools we shall use are the locally constrained inverse curvature flows in hyperbolic space and in the sphere.

The Minkowski problem for the k-torsional rigidity

Fri, 30 May 2025 00:00:00 GMT

Xia Zhao, Peibiao Zhao — P. Salani [Adv. Math., 229 (2012)] introduced the k-torsional rigidity associated with a k-Hessian equation and obtained the Brunn-Minkowski inequalities w.r.t. the torsional rigidity in ℝ³. Following this work, we first construct, in the present paper, a Hadamard variational formula for the k-torsional rigidity with 1≤ k≤ n-1, then we can deduce a k-torsional measure from the Hadamard variational formula. Based on the k-torsional measure, we propose the Minkowski problem for the k-torsional rigidity and confirm the existence of its smooth non-even solutions by the method of a curvature flow. Specially, a new proof method for the uniform lower bound estimation in the C⁰ estimation for the solution to the curvature flow is presented with the help of invariant functional Φ(Ωₜ).

Upper bounds for the Alexandrov-Fenchel deficit via integral formulas

Thu, 20 Mar 2025 00:00:00 GMT

Kwok-Kun Kwong, Yong Wei — We derive a number of sharp upper bounds for the deficit in the Alexandrov-Fenchel inequality using a weighted Minkowski integral formula and an integral formula for the deficit in Jensen's inequality. Our estimates yield results under weaker convexity assumptions compared to approaches based on inverse curvature flows. The use of weighted formulas provides flexibility in deriving inequalities with different weight functions. Furthermore, our estimates are more quantitative as they include a distance term measuring the domain's deviation from a reference ball. We also analyze the stability of a weighted geometric inequality from a recent paper via analysis of the support function on the sphere and show that, with an optimal choice of the origin, this inequality is stronger than the classical isoperimetric inequality.

The dual Minkowski problem for q-torsional rigidity

Wed, 16 Oct 2024 00:00:00 GMT

Xia Zhao, Peibiao Zhao — The Minkowski problem for torsional rigidity (2-torsional rigidity) was firstly studied by Colesanti and Fimiani using variational method. Moreover, Hu also studied this problem by the method of curvature flows and obtained the existence of smooth even solutions. In addition, the smooth non-even solutions to the Orlicz Minkowski problem w. r. t q-torsional rigidity were given by Zhao et al. through a Gauss curvature flow. The dual curvature measure and the dual Minkowski problem were first posed and considered by Huang, Lutwak, Yang and Zhang in. The dual Minkowski problem is a very important problem, which has greatly contributed to the development of the dual Brunn-Minkowski theory and extended the other types dual Minkowski problem. To the best of our knowledge, the dual Minkowski problem w. r. t (q) torsional rigidity is still open because the dual (q) torsional measure is blank. Thus, it is a natural problem to consider the dual Minkowski problem for (q) torsional rigidity. In this paper, we introduce the p-th dual q-torsional measure and propose the p-th dual Minkowski problem for q-torsional rigidity with q>1. Then we confirm the existence of smooth even solutions for p

A fully nonlinear locally constrained curvature flow for capillary hypersurface

Sun, 25 Aug 2024 00:00:00 GMT

Xinqun Mei, Liangjun Weng — In this article, we study a locally constrained fully nonlinear curvature flow for convex capillary hypersurfaces in half-space. We prove that the flow preserves the convexity, exists for all time, and converges smoothly to a spherical cap. This can be viewed as the fully nonlinear counterpart of the result in. As a byproduct, a high-order capillary isoperimetric ratio (1.6) evolves monotonically along this flow, which yields a class of the Alexandrov-Fenchel inequalities.

New quermassintegral and Poincaré type inequalities for non-convex domains

Mon, 12 Aug 2024 00:00:00 GMT

Yingxiang Hu, Mohammad N. Ivaki — In the first part of this paper, we study the following non-homogeneous, locally constrained inverse curvature flow in Euclidean space ℝⁿ⁺¹, \beginalign* \dotx=\left(\frac1{\frac{E_k(\hatκ)}{E_k-1(\hatκ)}-α}-\langle x,ν\rangle\right)ν, \quad k=2,3,\ldots,n-1. \endalign* Assuming that the initial hypersurface 𝓜₀ ⊂ ℝⁿ⁺¹ is star-shaped and its shifted principal curvatures hatκ=κ+α(1,…,1) lie in the convex set \beginalign* Γ_α,k:=Γ_k-1\cap \{λ\in \mathbbR^n:\, E_k(λ)-αE_k-1(λ)>0\}, \endalign* we show that the flow admits a smooth solution that exists for all positive times, and it converges smoothly to a round sphere. As a corollary, we obtain a new set of Alexandrov-Fenchel-type inequalities for non-convex domains. In the second part, we derive a Poincaré type inequality for k-convex hypersurfaces which complements a more general version of the well-known Heintze-Karcher inequality.

A Comprehensive Review of Solitonic Inequalities in Riemannian Geometry

Fri, 09 Aug 2024 00:00:00 GMT

Bang-Yen Chen, Majid Ali Choudhary, Mohammed Nisar, Mohd Danish Siddiqi — In Riemannian geometry, Ricci soliton inequalities are an important field of study that provide profound insights into the geometric and analytic characteristics of Riemannian manifolds. An extensive study of Ricci soliton inequalities is given in this review article, which also summarizes their historical evolution, core ideas, important findings, and applications. We investigate the complex interactions between curvature conditions and geometric inequalities as well as the several kinds of Ricci solitons, such as expanding, steady, and shrinking solitons. We also go over current developments, unresolved issues, and possible paths for further study in this fascinating area.

Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space

Tue, 11 Jun 2024 00:00:00 GMT

Xinqun Mei, Liangjun Weng — In this article, we first introduce the quermassintegrals for compact hypersurfaces with capillary boundaries in hyperbolic space from a variational viewpoint, and then we solve an isoperimetric type problem in hyperbolic space. By constructing a new locally constrained inverse curvature flow, we obtain the Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space. This generalizes a theorem of Brendle-Guan-Li for convex closed hypersurfaces in hyperbolic space.

A new trick for for Santalo's integral geometric proof of the strong Bonnesen inequality

Mon, 03 Jun 2024 00:00:00 GMT

Michael E. Gage — Given a lamina K whose boundary ∂K is convex we define the Bonnesen functional by integrating over the position and orientation of a disk of radius r its intersections with the lamina and its boundary. B(r) = 1/2π∫ (n/2-ν)dx dy dθ=rL-A-πr² where n is the number of intersections of the boundary of K with the boundary of the disk and ν (with values either 0 or 1) is the number of intersections of the interiors. Analyzing the interval on which B(r)≥ 0 leads to a number of isoperimetric inequalities. For example Santalo observed that B(r) is non-negative on [rᵢₙ,rₒᵤₜ] where rᵢₙ, the inradius, is the largest disk contained in K and rₒᵤₜ, the outradius, is the smallest disk containing K. For this configuration if the bodies intersect than their boundaries must intersect in 2 or more points (generically) so the integrand is always non-negative and B(r)≥ 0. In this paper we show that B(r)≥ 0 on the interval[ρᵢₙ, ρₒᵤₜ] where these are the inner and outer radii of the annulus of minimal width which surrounds ∂K. In this case the integrand is not always positive but we carefully analyze the integral and use an "averaging trick" to balance regions where the integrand is negative with regions where the integrand is positive to show that B(r)≥ 0 The final isoperimetric inequalities obtained are not new, the same results have been obtained by cleverly cutting ∂K and doubling it to create two centrally symmetric curves and analyzing those, but I believe that the averaging trick is new in this context. The paper also reviews the concept of "positive centers" and shows how the isoperimetric inequalities can be used to the show that the final shape of the curve shortening flow in Euclidean and Minkowski geometries are respectively the disk and the isoperimetrix.

Neckpinch — Geometric inequalities