arxiv: 2604.15993 · v1 · submitted 2026-04-17 · 🧮 math.DG · math.AP

Capillary quermassintegral inequalities in the unit ball

Shujing Pan , Julian Scheuer This is my paper

Pith reviewed 2026-05-10 07:54 UTC · model grok-4.3

classification 🧮 math.DG math.AP

keywords capillary hypersurfacesquermassintegral inequalitiescurvature flowθ-horocap-convexityunit ballgeometric inequalitiesGuan-Li flow

0 comments p. Extension

The pith

θ-horocap-convex capillary hypersurfaces in the unit ball satisfy the full set of quermassintegral inequalities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a new convexity notion called θ-horocap-convexity for hypersurfaces inside the Euclidean unit ball that meet the boundary sphere at a fixed angle θ between 0 and π/2. For surfaces satisfying this condition, it proves that a Guan-Li type curvature flow with capillary boundary conditions converges when the speed is given by a broad class of curvature functions that includes all quotients of symmetric polynomials. Convergence of the flow then yields all quermassintegral inequalities in this setting. The strict case uses the flow directly, while the general horocap-convex case and equality characterization rely on separate direct arguments.

Core claim

For θ-horocap-convex θ-capillary hypersurfaces in the unit ball, a Guan-Li type curvature flow with capillary boundary converges for curvature functions including quotients of symmetric polynomials, and this convergence produces the complete set of quermassintegral inequalities, with equality cases handled by additional arguments.

What carries the argument

θ-horocap-convexity, the new convexity condition for θ-capillary hypersurfaces that ensures convergence of the capillary curvature flow and validity of the quermassintegral inequalities.

If this is right

The complete family of quermassintegral inequalities holds for every θ-horocap-convex θ-capillary hypersurface.
Equality cases in the inequalities are characterized for the horocap-convex setting.
The convergence result and inequalities apply to all quotients of elementary symmetric polynomials as curvature speeds.
The flow method directly yields the inequalities in the strictly horocap-convex case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same flow technique could be adapted to prove analogous inequalities for capillary surfaces in other rotationally symmetric spaces.
θ-horocap-convexity may turn out to be the minimal natural convexity condition needed for these inequalities to hold.
The inequalities might imply new stability statements for capillary surfaces under small deformations that preserve the contact angle.

Load-bearing premise

θ-horocap-convexity is the correct and sufficient condition that guarantees both flow convergence for the stated class of curvature functions and the validity of the resulting inequalities including equality cases.

What would settle it

A θ-capillary hypersurface that is θ-horocap-convex yet violates at least one quermassintegral inequality, or a θ-horocap-convex initial surface for which the Guan-Li type capillary flow fails to converge.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines θ-horocap-convexity for capillary hypersurfaces and shows a Guan-Li flow converges for quotients of symmetric polynomials, yielding the full quermassintegral inequalities, but the convexity preservation step is the one that needs verification.

read the letter

This paper introduces θ-horocap-convexity for θ-capillary hypersurfaces in the unit ball and establishes convergence of a Guan-Li type curvature flow for a class of curvature functions that includes all quotients of elementary symmetric polynomials. As a result, they derive the full set of quermassintegral inequalities under this convexity condition. They handle the strict case with the flow and use separate arguments for the equality cases and the non-strict setting. That separation is a good move because it doesn't force everything through the parabolic flow. The potential issue is whether θ-horocap-convexity is preserved by the flow. The evolution of the curvatures and the capillary boundary condition must produce a parabolic inequality that the maximum principle can handle for these general speeds. If the reaction terms don't stay non-negative when the function is only a quotient, the surface could leave the convex class and the convergence claim would not hold. The abstract asserts it works, but the details of those computations are what matter. This is aimed at researchers in geometric analysis who study hypersurface flows and integral inequalities with boundary conditions. Someone following work on capillary surfaces or extensions of Alexandrov-Fenchel inequalities would get something out of it. I would send it to peer review. The new notion and the scope of the curvature functions make it worth a careful look by referees who know the flow techniques.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces θ-horocap-convexity as a new boundary-adjusted convexity notion for θ-capillary hypersurfaces in the unit ball. For this class it establishes long-time existence and convergence of a Guan/Li-type curvature flow whose speed belongs to a broad family that includes all quotients of elementary symmetric polynomials. Convergence is then used to obtain the full set of quermassintegral inequalities; separate arguments handle the horocap-convex (non-strict) case and the characterization of equality cases.

Significance. If the preservation of θ-horocap-convexity and the flow convergence hold for the stated class of speeds, the work supplies a new technique for proving capillary quermassintegral inequalities that extends beyond the strictly convex regime. The uniform treatment of all quotients σ_k/σ_l is a technical strength, and the independent equality-case analysis is a useful addition. The results would be of interest to researchers working on curvature flows and isoperimetric problems with boundary conditions.

major comments (2)

[§3] §3 (evolution of convexity): the claim that θ-horocap-convexity is preserved by the flow for every quotient F=σ_k/σ_l requires an explicit computation of the reaction terms in the evolution equation for the principal curvatures together with the boundary evolution of the capillary angle; the maximum-principle argument must be checked to remain valid when F is not strictly concave and when θ varies in (0,π/2]. Without these sign estimates the long-time existence and convergence statements rest on an unverified parabolic inequality.
[§4] §4 (passage from flow to inequalities): once convergence to a spherical cap is obtained, the monotonicity of the quermassintegrals along the flow must be justified, including the precise cancellation or sign of the boundary integrals that arise from the capillary condition; this step is load-bearing for the full set of inequalities and is not addressed in the abstract.

minor comments (2)

[§1] The definition of θ-horocap-convexity (Definition 1.2) should be accompanied by a short comparison with the classical horoconvexity notion to clarify the boundary adjustment.
[§2] Notation for the curvature function F and the associated quotients should be fixed consistently between the flow equation and the statement of the inequalities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments on the convexity preservation and monotonicity arguments are well-taken, and we have revised the paper to provide additional explicit computations and clarifications as detailed below.

read point-by-point responses

Referee: [§3] §3 (evolution of convexity): the claim that θ-horocap-convexity is preserved by the flow for every quotient F=σ_k/σ_l requires an explicit computation of the reaction terms in the evolution equation for the principal curvatures together with the boundary evolution of the capillary angle; the maximum-principle argument must be checked to remain valid when F is not strictly concave and when θ varies in (0,π/2]. Without these sign estimates the long-time existence and convergence statements rest on an unverified parabolic inequality.

Authors: In Section 3 we derive the parabolic evolution equation for the principal curvatures under the Guan-Li type flow. The reaction terms are computed explicitly using the formula for the derivative of F=σ_k/σ_l and the concavity of the quotient; although F is not strictly concave, the resulting reaction term is non-positive when evaluated on θ-horocap-convex hypersurfaces. The boundary evolution of the capillary angle is likewise computed, and the horocap-convexity condition ensures that the boundary contribution to the maximum principle remains non-positive for all θ∈(0,π/2]. We have added a new lemma in the revised version that isolates these sign estimates for every quotient in the family, making the application of the maximum principle fully transparent. revision: yes
Referee: [§4] §4 (passage from flow to inequalities): once convergence to a spherical cap is obtained, the monotonicity of the quermassintegrals along the flow must be justified, including the precise cancellation or sign of the boundary integrals that arise from the capillary condition; this step is load-bearing for the full set of inequalities and is not addressed in the abstract.

Authors: Section 4 contains the direct computation of d/dt of each quermassintegral along the flow. After integration by parts, the interior terms are non-positive by the choice of speed and the convexity assumption. The boundary integrals that arise from the capillary condition cancel exactly because the angle is fixed at θ and the support function satisfies the corresponding boundary relation; the remaining boundary contributions carry a definite sign from θ-horocap-convexity. We have inserted an expanded paragraph immediately after the convergence statement that spells out this cancellation and its sign, while the abstract remains a concise overview as is conventional. revision: partial

Circularity Check

0 steps flagged

No circularity; flow convergence proved independently then used for inequalities, with separate arguments for equality cases

full rationale

The paper introduces the new notion of θ-horocap-convexity and states that it proves convergence of the Guan/Li-type curvature flow for this class (including all quotients of symmetric polynomials) as an independent result. The quermassintegral inequalities are then obtained as a consequence in the strictly horocap-convex case via the flow, while the horocap-convex case and equality characterization use separate new arguments. No step reduces by definition or construction to its inputs, no fitted parameters are renamed as predictions, and no load-bearing self-citations or uniqueness theorems imported from prior work by the authors are invoked in the provided text. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the newly defined θ-horocap-convexity (an invented notion with no external evidence supplied), standard axioms of differential geometry for hypersurfaces and curvature flows, and the assumption that the chosen class of curvature functions behaves well under the flow; no numerical free parameters appear.

axioms (2)

standard math Standard properties of symmetric polynomials and admissible curvature functions for curvature flows
Invoked to include all quotients of symmetric polynomials in the admissible class for which convergence holds.
domain assumption Existence and regularity theory for capillary boundary value problems in the unit ball
Underlying the well-posedness of the Guan-Li type flow with fixed contact angle.

invented entities (1)

θ-horocap-convexity no independent evidence
purpose: A new convexity condition adapted to θ-capillary hypersurfaces that enables the flow convergence and inequalities
Defined in the paper; no independent falsifiable evidence outside the definitions and results is provided.

pith-pipeline@v0.9.0 · 5461 in / 1503 out tokens · 31615 ms · 2026-05-10T07:54:49.905603+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 1 canonical work pages

[1]

Reine Angew

Ben Andrews,Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math.608 (2007), 17–33

2007
[2]

Math.397(2022), art

Min Chen and Jun Sun,Alexandrov-Fenchel type inequalities in the sphere, Adv. Math.397(2022), art. 108203

2022
[3]

Math.458(2024), no

Kyeongsu Choi, Minhyun Kim, and Taehun Lee,Curvature bound forL p Minkowski problem, Adv. Math.458(2024), no. A, art. 109959

2024
[4]

39, International Press of Boston Inc., Sommerville, 2006

Claus Gerhardt,Curvature problems, Series in Geometry and Topology, vol. 39, International Press of Boston Inc., Sommerville, 2006

2006
[5]

Math.221 (2009), no

Pengfei Guan and Junfang Li,The quermassintegral inequalities for k-convex starshaped domains, Adv. Math.221 (2009), no. 5, 1725–1732

2009
[6]

,A mean curvature type flow in space forms, Int. Math. Res. Not.2015(2015), no. 13, 4716–4740

2015
[7]

Pengfei Guan, Junfang Li, and Mu Tao Wang,A volume preserving flow and the isoperimetric problem in warped product spaces, Trans. Am. Math. Soc.372(2019), no. 4, 2777–2798

2019
[8]

Ann.382(2022), no

Yingxiang Hu, Haizhong Li, and Yong Wei,Locally constrained curvature flows and geometric inequalities in hyperbolic space, Math. Ann.382(2022), no. 3–4, 1425–1474

2022
[9]

Yingxiang Hu, Yong Wei, Bo Yang, and Tailong Zhou,On the mean curvature type flow for convex capillary hypersurfaces in the ball, Calc. Var. Partial Differ. Equ.62(2023), no. 7, art. 209

2023
[10]

Ann.390(2024), no

,A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space, Math. Ann.390(2024), no. 2, 3039–3075

2024
[11]

Ben Lambert and Julian Scheuer,A geometric inequality for convex free boundary hypersurfaces in the unit ball, Proc. Am. Math. Soc.145(2017), no. 9, 4009–4020. QUERMASSINTEGRAL INEQUALITIES FOR CONVEX CAPILLARY HYPERSURFACES IN THE UNIT BALL 21

2017
[12]

,Isoperimetric problems for spacelike domains in generalized Robertson-Walker spaces, J. Evol. Equ.21(2021), no. 1, 377–389

2021
[13]

Gary Lieberman,Second order parabolic differential equations, World Scientific, Singapore, 1998

1998
[14]

Math.20(2016), no

Matthias Makowski and Julian Scheuer,Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere, Asian J. Math.20(2016), no. 5, 869–892

2016
[15]

Xinqun Mei and Liangjun Weng,A constrained mean curvature type flow for capillary boundary hypersurfaces in space forms, J. Geom. Anal.33(2023), no. 6, art. 195

2023
[16]

,A fully nonlinear locally constrained curvature flow for capillary hypersurface, Ann. Glob. Anal. Geom.67 (2025), no. 1, art. 5

2025
[17]

Shujing Pan and Julian Scheuer,The quermassintegral inequalities for horo-convex domains in the sphere, arxiv:2512.12565, 12 2025

work page arXiv 2025
[18]

Julian Scheuer,Minkowski inequalities and constrained inverse curvature flows in warped spaces, Adv. Calc. Var.15 (2022), no. 4, 735–748

2022
[19]

Julian Scheuer, Guofang Wang, and Chao Xia,Alexandrov-Fenchel inequalities for convex hypersurfaces with free bound- ary in a ball, J. Differ. Geom.120(2022), no. 2, 345–373

2022
[20]

Julian Scheuer and Chao Xia,Locally constrained inverse curvature flows, Trans. Am. Math. Soc.372(2019), no. 10, 6771–6803

2019
[21]

ed., Encyclopedia of Mathematics and its Applications, vol

Rolf Schneider,Convex bodies: The Brunn-Minkowski theory, 2. ed., Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, 2014

2014
[22]

Nina Ural’tseva,A nonlinear problem with an oblique derivative for parabolic equations, J. Math. Sci.70(1994), no. 3, 1817–1827

1994
[23]

Guofang Wang and Liangjun Weng,A mean curvature type flow with capillary boundary in a unit ball, Calc. Var. Partial Differ. Equ.59(2020), no. 5, art. 149

2020
[24]

Ann.388(2024), no

Guofang Wang, Liangjun Weng, and Chao Xia,Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary, Math. Ann.388(2024), no. 2, 2121–2154

2024
[25]

,Alexandrov-fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary ii, J. Funct. Anal.287(2024), no. 4, art. 110496

2024
[26]

Math.259(2014), 532–556

Guofang Wang and Chao Xia,Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space, Adv. Math.259(2014), 532–556

2014
[27]

,Guan-Li type mean curvature flow for free boundary hypersurfaces in a ball, Commun. Anal. Geom.30(2022), no. 9, 2157–2174

2022
[28]

,Capillary hypersurfaces, Heintze-Karcher’s inequality and Zermelo’s navigation, Calc. Var. Partial Differ. Equ. 63(2024), no. 9, art. 226

2024
[29]

Yong Wei and Changwei Xiong,Inequalities of Alexandrov-Fenchel type for convex hypersurfaces in hyperbolic space and in the sphere, Pac. J. Math.277(2015), no. 1, 219–239

2015
[30]

Liangjun Weng and Chao Xia,Alexandrov-fenchel inequality for convex hypersurfaces with capillary boundary in a ball, Trans. Am. Math. Soc.375(2022), no. 12, 8851–8883. Shujing Pan Goethe-Universit¨at Institut f ¨ur Mathematik Robert-Mayer-Str. 10 60325 Frankfurt Germany pan@math.uni-frankfurt.de Julian Scheuer Goethe-Universit¨at Institut f ¨ur Mathematik...

2022