Quermassintegral Inequalities for Horo-Convex Hypersurfaces

• The paper presents sharp quermassintegral inequalities for horo-convex hypersurfaces using curvature flows that ensure exponential convergence to geodesic spheres.
• It employs locally constrained curvature flows and Minkowski identities to derive stringent bounds and rigidity characterizations in hyperbolic space.
• The results extend classical Alexandrov–Fenchel inequalities by offering quantitative stability estimates and opening pathways for generalizations in geometric analysis.

Quermassintegral inequalities for horo-convex hypersurfaces in hyperbolic space provide a geometric framework generalizing classical Alexandrov–Fenchel inequalities. These results yield sharp bounds for curvature integrals and quermassintegrals — both in their classical and modified forms — with rigidity characterizations for equality cases. The underlying methodology leverages locally constrained curvature flows preserving specific quermassintegrals and exploits the horo-convexity structure to establish exponential convergence to geodesic spheres. This theory has undergone extensive development, notably in the foundational works of Ge–Wang–Wu, Andrews–Chen–Wei, Hu–Li–Wei, and recent stability analyses, thereby refining the class of admissible hypersurfaces and expanding the inequalities’ scope.

1. Fundamental Concepts and Definitions

Hyperbolic space $\mathbb{H}^{n+1}$ is frequently represented as the forward unit hyperboloid in Minkowski $\mathbb{R}^{n+2}$ or via warped-product coordinates $([0,\infty)\times S^n,\, dr^2+\sinh^2 r\,g_{S^n})$ (Andrews et al., 2018, Hu et al., 2020, Gao et al., 18 Jan 2024).

A horosphere is a hypersurface of constant principal curvatures all equal to $1$. A domain $\Omega$ is (strictly) horo-convex if its boundary, a smooth closed hypersurface $M\subset\mathbb{H}^{n+1}$, satisfies $\kappa_i>1$ for all principal curvatures $\kappa_i$ at every point (Andrews et al., 2018, Andrews et al., 2017, Gao et al., 18 Jan 2024). The shifted principal curvatures are $\lambda_i=\kappa_i-1$, and for horo-convexity, $\lambda_i>0$ everywhere.

Quermassintegrals $W_k(\Omega)$ for $0\leq k\leq n$ are fundamental in integral geometry. For $k\ge1$,

$$W_k(\Omega) = \frac{1}{|M|}\int_M E_{k-1}(\kappa)\,d\mu,$$

where $E_{k-1}$ is the normalized elementary symmetric function. $W_0(\Omega)=|\Omega|$ is the volume. Modified quermassintegrals are defined as: $$\widetilde{W}_k(\Omega) = \sum_{i=0}^k (-1)^{k-i} \binom{k}{i} W_i(\Omega).$$ These quantities encode geometric invariants of convex domains and provide the natural setting for the inequalities addressed below.

2. Sharp Quermassintegral Inequalities for Horo-Convex Hypersurfaces

The main results generalize the classical Alexandrov–Fenchel inequalities to horo-convex hypersurfaces in hyperbolic space. For strictly horo-convex domains $\Omega\subset\mathbb{H}^{n+1}$ and any indices $0\leq l<k\leq n$, Andrews–Chen–Wei establish (Andrews et al., 2018, Hu et al., 2020, Gao et al., 18 Jan 2024): $$\widetilde{W}_k(\Omega) \geq \widetilde{f}_k\left(\widetilde{f}_l^{-1}(\widetilde{W}_l(\Omega))\right),$$ where $\widetilde{f}_k(r)=\widetilde{W}_k(B(r))$ for the geodesic ball $B(r)$ of radius $r$. Equality holds precisely when $\Omega$ is a geodesic ball. For $l=0$, this yields a linear combination of standard quermassintegrals: $$\sum_{i=0}^{k} (-1)^{k-i} \binom{k}{i} W_i(\Omega) \geq \widetilde{f}_k(\widetilde{f}_0^{-1}(W_0(\Omega))).$$

Odd-order quermassintegral inequalities, as in the work of Ge–Wang–Wu and Hu–Li, are sharp for horo-convex domains (Hu et al., 2018, Ge et al., 2013): $$W_{2k+1}(\Omega) \geq \frac{\omega_{n-1}}{n} \sum_{i=0}^k \frac{n-1-2k}{n-1-2i} \binom{k}{i} \left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{n-1-2i}{n-1}},$$ with equality only for geodesic spheres.

Even-order curvature integral inequalities, e.g., for the $k$-th mean curvature, are of the form (Ge et al., 2013, Ge et al., 2013): $$\int_\Sigma \sigma_{2k} \, d\mu \geq \binom{n-1}{2k} \omega_{n-1} \left[\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{1/k} + \left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{1}{k}\frac{n-1-2k}{n-1}}\right]^k.$$

3. Locally Constrained Curvature Flows: The Analytic Framework

The proof strategies are fundamentally analytic, using locally constrained curvature flows tailored to horo-convexity (Hu et al., 2020, Andrews et al., 2018, Gao et al., 18 Jan 2024, Andrews et al., 2017).

For the classical quermassintegral inequalities, one deploys a flow preserving a chosen quermassintegral: $\partial_t X = (F_m - u) \nu,$ where $F_m = E_m/E_{m-1}$ and $u$ is the support function. Preservation of $W_m$ and monotonicity of $W_{m+1}$ is guaranteed via the Minkowski identities and Newton–Maclaurin inequalities.

The modified quermassintegral inequalities utilize the shifted flow: $$\partial_t X = ((\cosh r-u)\widetilde{F}_{m-1} - u\widetilde{F}_m)\nu,$$ with $\widetilde{F}_m = E_m(\kappa)/E_{m-1}(\kappa)$ for shifted curvatures $\kappa_i = \lambda_i-1$. These flows preserve strict horo-convexity and exhibit exponential convergence to geodesic spheres, as proved via tensor maximum principles, Schauder theory, and spectral analysis.

4. Rigidity, Stability, and Equality Cases

Rigidity in these inequalities is absolute: equality forces the domain to be a geodesic ball (Andrews et al., 2018, Gao et al., 18 Jan 2024, Sahjwani et al., 2023). The strong maximum principle along the flow implies that if any monotone functional remains constant, the hypersurface remains totally umbilic for all time, precluding non-spherical solutions.

Recent analyses have yielded quantitative stability estimates: for a domain with quermassintegral deficit $\delta_{m+1}(\Omega)$,

$$d_H\left(\Omega,B_R\right) \leq C\left[\delta_{m+1}(\Omega)\right]^{\alpha}$$

with explicit exponent $\alpha=\frac{1}{m+2}$ (or $\frac{1}{3}$ under mean curvature bounds), where $d_H$ is the Hausdorff distance to a sphere and $C$ depends on dimension, inradius, and curvature quotient bounds (Sahjwani et al., 2023, Gao et al., 18 Jan 2024).

The proof leverages uniform initial-value independent curvature estimates for the flows and pulls back almost-umbilicity results from Euclidean ball geometry via conformal identifications.

5. Generalizations, Spherical Analogues, and Related Developments

The horo-convexity concept has a spherical analogue, recently formalized for unit spheres by Pan–Scheuer (Pan et al., 14 Dec 2025). This notion retains the essential property that the corresponding flow converges to geodesic spheres and yields the full family of quermassintegral inequalities for horo-convex hypersurfaces in the sphere, with rigidity and monotonicity properties closely paralleling the hyperbolic case.

Further generalizations include flows for domains with nonnegative sectional curvature, which strictly enlarge the admissible class beyond horo-convexity, and extensions to arbitrary warped product spaces with conformal Killing fields remain open (Hu et al., 2018, Pan et al., 14 Dec 2025).

6. Technical Tools and Auxiliary Results

The analysis employs variational formulas for modified quermassintegrals, evolution equations for curvature invariants, and Newton–Maclaurin inequalities in the Gårding cone $\Gamma_k^+$. Minkowski-type identities for shifted curvatures, divergence-free properties of higher-order Newton transforms, and interpolation inequalities complement the geometric flow machinery.

Additionally, conformal identifications with Euclidean convex hypersurfaces allow application of quantitative nearly-umbilical theorems to bridge stability estimates from hyperbolic to Euclidean settings (Gao et al., 18 Jan 2024, Sahjwani et al., 2023).

7. Significance, Impact, and Open Problems

Quermassintegral inequalities for horo-convex hypersurfaces provide sharp bounds for fundamental geometric quantities in hyperbolic and spherical geometries. They generalize classical integral-geometric inequalities, define explicit rigidity and stability thresholds, and furnish analytic tools for geometric analysis. Open questions concern weakening horo-convexity (e.g., $m$-convexity or star-shapedness), extension to other symmetric spaces, and the development of weak solution theories for non-smooth hypersurfaces (Ge et al., 2013, Pan et al., 14 Dec 2025).

These results form a cohesive and expanding framework, integrating curvature flows, rigidity theorems, and stability phenomena into the study of geometric inequalities in non-Euclidean background geometries.