Geometric inequalities

We study quermassintegral-preserving curvature flows in ${\mathbb{H}}^{n+1}$, $n\ge 3$, driven by the curvature quotient $(E_k/E_{\ell }{)}^{1/(k-\ell )}$. For $1\le \ell <k$ with $k+\ell \le n$, we prove that positive sectional curvature is preserved. Starting from a smooth closed strictly convex hypersurface with positive sectional curvature, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. As an application, we establish sharp Alexandrov–Fenchel inequalities between $W_k$ and $W_{\ell }$ for smooth bounded convex domains whose boundaries have nonnegative sectional curvature, whenever $0\le \ell <k\le n$ and $k+\ell \le n$. Equality holds if and only if the domain is a geodesic ball. Together with previously known cases, these inequalities give the full family of Alexandrov–Fenchel inequalities for this class of domains in ${\mathbb{H}}^6$. We also construct counterexamples showing that $k+\ell \le n$ is the optimal range for preservation of positive sectional curvature by these flows.

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 $$⟨X+AX,\nu ⟩>0$$ for some fixed skew-symmetric linear map $A$, where $X$ is the position vector and $\nu$ is the outward unit normal. The vector field $X+AX$ is conformal Killing, with dilation component $X$ and rotational component $AX$; 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.

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 $(1\le l<k\le n)$. Using this flow, we establish a full family of the quermassintegral inequalities for horo-convex capillary hypersurfaces in a horoball with contact angle $\theta \in (0,\pi /2]$. Moreover, we weaken the horo-convexity to convexity and obtain the quermassintegral inequalities for convex capillary hypersurfaces between the $n$-th and $l$-th quermassintegrals.

We establish the Alexandrov-Fenchel inequality and its modified version for arbitrary, possibly nonsmooth, horospherically convex bodies in hyperbolic space ${\mathbb{H}}^{n+1},n\ge 2$, 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 $C^{1,\alpha }\cap W^{2,k+1}$ 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 $0\le k\le n-1$, we construct weak flows preserving the $k$-th modified quermassintegral ${\tilde{W}}_k$ 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 ${\tilde{W}}_{k+1}$ 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.

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 $K_t$ starting from the boundary of a smooth bounded domain $\Omega \subset {\mathbb{R}}^{n+1}$, as well as for the integral Brakke flow $\{{\mu }_t\}$ with initial Radon measure ${\mu }_0={\mathcal{H}}^n└\partial \Omega$, 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 $P(\Omega )$ is the perimeter of $\Omega$ representing the area of $\partial \Omega$, and the equality holds if and only if $\Omega$ 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 $\Omega \subset {\mathbb{R}}^{n+1}$: \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 $\Omega$ is a round ball and the flow is standard smoothly shrinking round sphere.

We establish the eventual monotonicity of the Willmore deficit ${\int }_{\Sigma }{H}^{2}d\mu -16\pi$ along the mean curvature flow of smooth closed embedded strictly convex surfaces in ${\mathbb{R}}^3$, 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.

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.

This paper establishes sharp upper bounds for $p$-capacities ${\mathrm{Cap}}_{1<p<\infty }$ in the hyperbolic space ${\mathbb{H}}^n$ through hyperbolic quermassintegrals and effective curvature radii. The quermassintegral comparisons involve $W_{n-1}$, $W_{k+1}+k(n+1-k{)}^{-1}{W}_{k-1}$, and the pair $W_1\mid W_2$. For star-shaped, mean-convex or h-convex hypersurfaces, inverse mean curvature flow further produces curvature radii determined by $L^q$-averages of the normalized mean curvature and by moments of its squared hyperbolic excess. These radii convert the resulting estimates into sharp geodesic-ball comparisons for the capacity-to-area ratio. In the range $p>2m+1$, an interpolating radius combines the $2m$-th curvature-excess radius with the $L^{\infty }$ curvature scale, thereby linking the finite-moment and supremum regimes. Equality in the sharp comparisons characterizes geodesic balls.

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 $C^{\infty }$ convergence to the geodesic sphere centered at the fixed origin whose radius is determined by the preserved quermassintegral.

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 $\Sigma$ that satisfies $|\Sigma |\ge 4\pi q^2$ - where $q$ is the total charge - we show that the following inequality for the ADM mass $m$ holds: $m\ge \sqrt{\frac{|\Sigma |}{16\pi }}+q^2\sqrt{\frac{\pi }{|\Sigma |}}$, with equality if and only if $k\equiv 0$ 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 $\mathbf{P}$-inverse mean curvature flow and its weak formulation, which only depends on $(g,\mathbf{P})$ 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 ($k\equiv 0$), 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.

In this paper, we investigate the cocompact inverse ${\sigma }_k$ curvature flow in Minkowski space. We prove the longtime existence and convergence of this flow. As a consequence, quermassintegral inequalities are established.

In this work, we establish compactness and regularity results for complete gradient Laplacian solitons of closed $G_2$-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 $C^{1,\alpha }$ convergence. Our principal result shows that this $C^{1,\alpha }$ 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 $G_2$-solitons. Since the available $C^{1,\alpha }$ control does not directly close an elliptic bootstrap for the $G_2$-soliton, and no suitable pseudolocality theorem is available in this setting, we instead use the localised Perelman's functionals. These yield an entropy $\epsilon$-regularity theorem and a gap theorem for scalar-flat solitons. On the regular set, pointed $C^{1,\alpha }$ 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 $G_2$-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 $L^{\frac{7}{2}}$ -curvature bound then yields pointed $C^{\infty }$ compactness.

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 $H_n$-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.

This paper is about hypersurfaces with boundary lying in the Euclidean unit ball, which meet the unit sphere at a fixed angle $\theta \in (0,\frac{\pi }{2}]$. Such hypersurfaces are called $\theta$-capillary hypersurfaces and for those we introduce a new notion of convexity, which we call $\theta$-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 $\theta$-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.

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.

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.

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 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 $1\le k\le n-1$, 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 $\tau >0$ is a constant, $f$ is a positive smooth function defined on $S^{n-1}$ and ${\sigma }_{n-k}$ 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 $p<n-2$ by the method of a curvature flow which converges smoothly to the solution of equation (). Specially, a novel approach for the uniform lower bound estimation in the $C^0$ estimation for the solution to the curvature flow is presented with the help of invariant functional $\Phi ({\Omega }_t)$.

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 $\sqrt{3}$, with equality achieved only by the critical AdS-Schwarzschild metric. Consequently, Poincaré-Einstein fillings of $S^2\times S^1(\lambda )$ arising from these spaces have length parameter $\lambda \le (\sqrt{3}{)}^{-1}$, 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].

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.