Convex capillary hypersurfaces of prescribed curvature problem

Guofang Wang; Liangjun Weng; Xinqun Mei

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Convex capillary hypersurfaces of prescribed curvature problem

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Pith's one-line read This paper proves that every positive smooth capillary-even function on a spherical cap is realized as the k-th Weingarten curvature of a strictly convex capillary hypersurface in the upper half-space, for any 1 ≤ k ≤ n−1 and contact…

desk verdict A solid and significant extension of the Guan–Guan existence theory to capillary hypersurfaces, but the proof has a fixable C^0 gauge gap that needs to be addressed before the main theorem is fully trustworthy. read the letter →

arxiv 2504.14392 v1 pith:YAE2UYZK submitted 2025-04-19 math.DG math.AP

Xinqun Mei , Guofang Wang , Liangjun Weng This is my paper · ORCID

classification math.DGmath.AP MSC 53C2135B6535J6053C42

keywords capillaryhypersurfacesWeingartencurvatureHessianquotientequationRobinboundaryconditiondegreetheoryprescribedproblemsphericalcapsupportfunction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether a positive function prescribed on a spherical cap can be realized as the $k$-th Weingarten curvature of a strictly convex hypersurface that meets the boundary plane of the upper half-space at a fixed contact angle. The authors prove that the answer is yes for every smooth positive 'capillary even' function — one symmetric under horizontal reflection — for any $1 \le k \le n-1$ and contact angle $\theta \in (0, \pi/2]$. The proof rephrases the geometric problem as a Hessian quotient equation with a Robin boundary condition for the support function, then solves that equation by a priori estimates and degree theory. The result extends the capillary Minkowski problem ($k = n$) to all intermediate curvature functions, and also yields uniqueness near constant data and a counterexample showing that the classical volume-balance condition is not necessary for $k < n$.

What carries the argument

The load-bearing structure is the Hessian quotient equation (1.3), rewritten from the geometric prescription: $$\frac{\sigma_n(\$nabla^{2}$ h + h\$\sigma$)}{\sigma_{n-k}(\$nabla^{2}$ h + h\$\sigma$)} = $f^{{-1}}$ \quad \text{in } C_\$\theta$, \qquad \nabla_\mu h = \cot\$\theta$\, h \quad \text{on } \partial C_\$\theta$,$$ where $h$ is the capillary support function on the spherical cap $C_\theta$ and $\sigma$ is the round metric. The matrix $A = \nabla^2 h + h\sigma$ has eigenvalues equal to the principal radii of the hypersurface, so admissibility ($A>0$) is exactly strict convexity. The argument runs on three mechanisms: a maximum-principle bound on the capillary inner radius (Lemma 3.2); a barrier construction (Lemmas 3.4–3.5) showing $\max |\nabla^2 h| \le C(1+\|h\|_{C^0})$, the quantitative estimate that closes the $C^2$ bound; and a geometric lemma (Lemma 2.3) bounding $\rho_+(\hat\Sigma,\theta)^2/\rho_-(\hat\Sigma,\theta) \le C \max \lambda_n$, which converts inner-radius control into an outer-radius bound and hence a $C^0$ bound once translations are fixed by capillary evenness. Degree theory on the map $G(h,t) = \sigma_n(A)/\sigma_{n-k}(A) - f_t$ then yields existence, with the linearized operator at the constant solution $\ell$ having kernel exactly the horizontal translations, which the evenness condition kills.

What would settle it

A direct numerical or analytical continuation from $f\equiv1$ along a path of capillary even functions, monitoring the eigenvalues of $\nabla^2 h + h\sigma$; if a smooth positive even $f$ with bounded $C^3$ norm produced a support function leaving a fixed $C^{4,\gamma}$ ball or losing convexity, the a priori estimate of Theorem 3.1 and the degree-theoretic existence proof would be false.

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Extended reading notes

Core claim

The central discovery is an existence theorem (Theorem 1.1): for $\theta \in (0,\pi/2]$, $1 \le k \le n-1$, and any positive smooth capillary even function $f$ on the spherical cap $C_\theta$, there is a strictly convex capillary hypersurface $\Sigma \subset \mathbb{R}^{n+1}_+$ with $W_k(\tilde\nu^{-1}(\xi)) = f(\xi)$ for all $\xi \in C_\theta$. Here $\tilde\nu$ is the capillary Gauss map and $W_k$ is the $k$-th elementary symmetric function of the principal curvatures. The problem is equivalent to solving $\sigma_n(\nabla^2 h + h\sigma)/\sigma_{n-k}(\nabla^2 h + h\sigma) = f^{-1}$ on $C_\theta$ with Robin boundary condition $\nabla_\mu h = \cot\theta\, h$, where $h$ is the capillary support function. The main technical work is a set of a priori estimates: a bound on the capillary inner radius, a quantitative linear control of $|\nabla^2 h|$ by $\|h\|_{C^0}$, and a capillary version of a geometric lemma relating the capillary outer and inner radii to the maximal principal radius. These combine into a uniform $C^{4,\gamma}$ bound that feeds a degree-theoretic existence argument, whose base case is the unique constant-curvature capillary hypersurface. The paper also proves uniqueness for data close to constant (Theorem 1.2) and constructs a one-parameter family of capillary hypersurfaces for which $\int_{C_\theta} \frac{\langle \xi, E_\alpha\rangle}{W_k}\, dA_\sigma \ne 0$, showing that condition (1.2) is not necessary for $k < n$.

Load-bearing premise

Everything rests on the capillary evenness of the prescribed function — the symmetry that pins down horizontal translations, without which the a priori bounds driving the proof can fail.

Editorial extensions

If this is right

For any positive smooth capillary even $f$, the prescribed curvature equation (1.1) has a solution, so the full range $1\le k\le n-1$ of Weingarten curvatures is now covered in the capillary setting, not only $k=n$ (the Minkowski case).
At $\theta = \pi/2$, reflecting the solution across the boundary plane gives an alternative proof of the classical prescribed Weingarten curvature problem for closed convex hypersurfaces with even symmetry.
The a priori estimate of Theorem 3.1 is quantitative: the $C^{4,\gamma}$ norm of the support function is controlled by $n$, $k$, $\gamma$, $\min f$, and $\|f\|_{C^3}$ alone, so the solution family is compact over bounded sets of data.
Near constant data the solution is unique (Theorem 1.2), so there is a well-posed branch of solutions emanating from the unique constant-curvature capillary cap.
The integral condition (1.2), necessary and sufficient for the capillary Minkowski problem, is not necessary for $1\le k\le n-1$: the constructed one-parameter family of capillary hypersurfaces violates it.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof uses only that capillary evenness kills the kernel of the linearized operator at the constant solution; any reflection or finite symmetry group of $C_\theta$ with no nonzero invariant first harmonics would plausibly work the same way, so the symmetry assumption may be more convenient than essential.
The authors expect the theorem to hold for $\theta>\pi/2$; if the geometric radius-ratio bound and the boundary barriers can be extended to obtuse contact angles, the identical degree argument would go through, which is a clean test of the method.
The quantitative a priori estimates suggest a numerical continuation from the constant solution $h=\ell$ along a path of capillary even functions $f$, with uniform step control; the existence could then be exhibited computationally for concrete data.
If evenness is removed, the natural next question is whether a suitably normalized version of condition (1.2) becomes sufficient; the counterexample in Theorem 1.3 shows the raw condition is not necessary, so a sharper obstruction would be needed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper proves a natural open case: existence of strictly convex capillary hypersurfaces in the half-space with prescribed k-th Weingarten curvature, for 1 ≤ k ≤ n−1 and capillary even data. If the main theorem holds, it extends Guan-Guan's closed case and the authors' own capillary Minkowski result. The proof adapts known machinery—Chou-Wang, Lieberman–Trudinger, degree theory—and includes a genuinely new quantitative estimate: max principal radius is controlled linearly by the C^0 norm of the support function. That estimate is a real improvement over their earlier Minkowski paper.

The central a priori estimate (Theorem 3.1) is where the paper is currently softest. The proof derives ρ+^2 ≤ C ρ− max λ_n and then writes ≤ C(1+ρ+), which implicitly requires ||h||_{C0} ≤ C(1+ρ+). But horizontal translations are exact symmetries of the equation and of the capillary radii, so that bound is false without fixing the translation gauge. Evenness of h is the intended gauge, but no lemma states or proves that evenness forces the optimal cap center to the origin, which is what would give the missing C^0 bound. This is a load-bearing gap in the written proof. It looks fixable with a short argument, but as it stands Theorem 3.1 doesn't close.

There are also smaller issues. Lemma 2.3 asserts without proof that the maximal John ellipsoid touches the hypersurface at an interior point. The θ = π/2 endpoint is excluded from Theorem 3.1 although Theorem 1.1 claims it. And in §5 the expansion (5.2) has wrong coefficients—a_1 should be kH_1, and the H_2 term in a_2 is off. The counterexample's conclusion survives because the wrong terms integrate to zero by (5.1); so this is a cosmetic blemish, not a fatal one. The introduction also misstates Theorem 1.3: the constructed hypersurfaces show condition (1.2) is not necessary for solvability, not not sufficient.

Overall, the main theorem is plausible and important, the estimates are mostly detailed and correct, and none of the gaps look deep. But the C^0 gauge issue needs to be fixed before the proof is fully trustworthy. I'd send this to a serious referee, and I'd expect a revision that adds the missing gauge lemma and cleans up the endpoint and counterexample details.

Referee Report

3 major / 3 minor

Summary. The paper treats the prescribed k-th Weingarten curvature problem for strictly convex capillary hypersurfaces in the upper half-space R^{n+1}_+. It reformulates the problem as the Hessian quotient equation with Robin boundary condition (1.3) for the capillary support function h on Cθ, proves a priori C^{4,γ} estimates for admissible capillary even solutions (Theorem 3.1), and uses degree theory to obtain existence (Theorem 1.1), uniqueness near the constant solution (Theorem 1.2), and a counterexample showing that the capillary Minkowski condition (1.2) is not sufficient for 1≤k≤n−1 (Theorem 1.3). The main technical novelty is a quantitative bound relating the largest principal radius to the C0 norm of h, combined with a capillary version of Chou–Wang's geometric lemma.

Significance. The result is a natural and valuable extension of Guan–Guan's theorem for closed hypersurfaces and of the authors' capillary Minkowski result. The PDE formulation is clean, the degree-theoretic strategy is appropriate, and the paper correctly identifies capillary evenness as the translation-gauge condition. If the missing gauge argument is supplied, Theorem 1.1 would be a substantial contribution. The uniqueness theorem and the counterexample also add value. I am not convinced at present that Theorem 3.1 is proved as written, because the C0 estimate contains a load-bearing gap; the remaining issues are local and fixable.

major comments (3)

[§3, proof of Theorem 3.1] The conclusion ρ_+(Σhat,θ)≤C does not follow from the displayed chain. The text writes ρ_+(Σhat,θ)^2 ≤ Cρ_-(Σhat,θ) max λ_n ≤ C(1+ρ_+(Σhat,θ)), but Theorem 3.6 supplies max λ_n ≤ C(1+‖h‖_{C0}), not C(1+ρ_+). This matters because horizontal translations are exact symmetries of (1.3): h_a(ξ)=h(ξ)+Σ_α a_α⟨ξ,E_α⟩ solves the same equation and the same Robin condition, while the capillary radii ρ_±(Σhat,θ) are unchanged. Thus capillary radii alone cannot control ‖h‖_{C0} without fixing the translation gauge. Capillary evenness is exactly the assumption that should fix the gauge, but no lemma in Section 3 states or proves that evenness yields ‖h‖_{C0}≤C(1+ρ_+), for instance by showing the body is contained in a fixed ball centered on the vertical axis. Without such a lemma, the a priori estimate and hence the degree-theoretic proof of Theorem 1.1 are incomplete.
[§2.2, proof of Lemma 2.3] The existence of a contact point of ∂E_hatb with Σhat that is not on the flat boundary is asserted rather than proved. The sentence "otherwise all the touch points lie in ∂R^{n+1}, and it is impossible" does not explain why a maximizing ellipsoid in the capillary setting cannot have all its contact points on the supporting hyperplane. This point is needed for the inequality λ_{x,Σ}≥λ_{x,∂E_hatb} in (2.3) and therefore for the key estimate (1.6). Please provide a complete argument.
[Theorem 1.1 vs. Theorem 3.1] Theorem 3.1 and Lemmas 3.4–3.5 are stated only for θ∈(0,π/2), while Theorem 1.1 includes θ=π/2. The proof of Theorem 1.1 invokes Theorem 3.1 without explaining how the endpoint is handled. If the reflection reduction mentioned in the introduction is intended to cover θ=π/2, it should be stated explicitly in the proof of Theorem 1.1; otherwise a separate or limiting argument is needed.

minor comments (3)

[§5, Eq. (5.2)] The displayed expansion is algebraically incorrect. For A_t=I+tV, the first-order coefficient is kH_1, not (n−k)H_1, and the H_2 coefficient in a_2 should be (n−k)(2n−k−1)/2 instead of (n−k)(n+k−1)/2. The conclusion of Theorem 1.3 is unaffected because the erroneous terms multiply functions whose first moment against ⟨ξ,E_α⟩ vanishes by (5.1), but the formula should be corrected.
[§3, Theorem 3.6] The displayed lower bound c_0(1+‖h‖_{C0})^{n−1}σ appears to have the wrong sign in the exponent. The determinant lower bound σ_n(A)≥c together with the upper bound λ_i≤C(1+‖h‖_{C0}) gives λ_i≥c(1+‖h‖_{C0})^{-(n−1)}, so the exponent should be −(n−1). Please correct.
[§4, definition of B] The set B in the degree argument does not explicitly impose h>0, although Theorem 3.1 is stated for positive h. Please either include positivity in the definition of B or add a remark explaining that positivity follows from admissibility and the Robin condition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the existence proof is a genuine PDE/degree-theory derivation from prior independent geometric identities and uniqueness theorems.

full rationale

The central claim (Theorem 1.1) is proved by converting the capillary Weingarten problem into the Robin boundary value problem (1.3), establishing a priori estimates, and applying degree theory. None of these steps is equivalent to the statement being proved. The reduction to (1.3) is imported from [40, Proposition 2.4] and [41]; these are prior geometric support-function identities whose assumptions (strict convexity, constant contact angle) do not include the prescribed-curvature existence for general f. The degree computation uses [28, Corollary 1.2] for uniqueness of the constant-curvature capillary hypersurface; although G. Wang is a coauthor of both papers, this is a parameter-free external theorem, not a restatement of Theorem 1.1, and it does not determine the solution for the variable f. Capillary evenness (Definition 1.1) is an assumption on the data and on the admissible solution space that fixes the horizontal translation gauge; the paper explicitly acknowledges that removing it is the main open obstacle, which is a limitation rather than a disguised input. The proof does contain a potentially serious gap: in the proof of Theorem 3.1 the bound on the capillary outer radius is combined with Theorem 3.6 to conclude ||h||_{C0} <= C without an explicit lemma showing capillary evenness controls the horizontal translation of the support function. That is a correctness risk, as is the algebraic error in the expansion (5.2) used for Theorem 1.3 (a1 should be kH1, not (n-k)H1). These are mathematical gaps, not circular reductions; no fitted parameter is renamed as a prediction, and no self-citation is doing the work of the theorem's conclusion. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted to data; the constants in the estimates are universal. The proof imports several substantial background results, chiefly the uniqueness of constant capillary hypersurfaces and the geometric reduction from the authors' earlier work. No new entities are postulated.

assumptions (7)

domain assumption Uniqueness of constant k-th Weingarten capillary hypersurfaces in a wedge ([28, Corollary 1.2])
Used in the degree theory proof of Theorem 1.1 to identify the unique solution for f = 1 and to compute the degree as ±1 (Section 4).
domain assumption Equivalence between the geometric problem and the Hessian quotient equation with Robin boundary condition ([40, Proposition 2.4])
The whole PDE formulation (1.3) is taken from the authors' earlier work; not reproved here (Section 1.3).
domain assumption Chou-Wang geometric lemma adapted to capillary setting (Lemma 2.3)
Provides the key inequality ρ_+^2/ρ_- ≤ C(n) max λ_n; the proof invokes John's ellipsoid theorem and an unproved assertion about an interior contact point.
standard math Lieberman-Trudinger regularity theory for oblique derivative problems ([34, Theorem 1.1])
Used to pass from C^2 to C^{4,γ} in Theorem 3.1.
standard math Degree theory for fully nonlinear elliptic operators with oblique boundary conditions ([31,32])
Underpins the existence argument in Section 4.
standard math Properties of elementary symmetric functions and the Gårding cone (Propositions 2.1, 2.2)
Used throughout the estimates; cited from [33,52].
standard math John's ellipsoid theorem
Used in Lemma 2.3 to find the ellipsoid E with E ⊆ Σ̂ ⊆ nE.

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Pith. "Pith review of Convex capillary hypersurfaces of prescribed curvature problem." pith.science (2026). https://pith.science/paper/YAE2UYZK

@misc{pith2026250414392,
  author       = {Pith},
  title        = {Pith review of: Convex capillary hypersurfaces of prescribed curvature problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YAE2UYZK}},
  note         = {Machine review of arXiv:2504.14392}
}

abstract

In this paper, we study the prescribed $k$-th Weingarten curvature problem for convex capillary hypersurfaces in $\overline{\mathbb{R}^{n+1}_+}$. This problem naturally extends the prescribed $k$-th Weingarten curvature problem for closed convex hypersurfaces, previously investigated by Guan-Guan in [19], to the capillary setting. We reformulate the problem as the solvability of a Hessian quotient equation with a Robin boundary condition on a spherical cap. Under a natural sufficient condition, we establish the existence of a strictly convex capillary hypersurface with the prescribed $k$-th Weingarten curvature. This also extends our recent work on the capillary Minkowski problem in [40].

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Cited by 2 Pith papers

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