arXiv subjects

Guofang Wang

Publications and source records attributed to Guofang Wang.

At least 19 recordsLinked to original sources

arXiv2026

Optimal geometric inequalities and fully nonlinear conformal flows

We establish sharp Sobolev-type geometric inequalities on $\mathbb{S}^n$ involving the total $\sigma_k$-curvatures $\int_{\mathbb{S}^n}\sigma_k(g)\,dv_g$. These results extend the optimal inequalities of Guan--Wang~\cite{GWDuke} from the cone $\mathcal{C}_k$ to the strictly larger cone $\mathcal{C}_{k-1}$, thereby enlarging the range of admissible conformal metrics. Our approach is variational and is implemented through a fully nonlinear conformal flow. Working in $\mathcal{C}_{k-1}$ introduces substantial analytic difficulties; in particular, one must obtain $C^2$ a priori estimates while simultaneously verifying that the flow remains parabolic. We resolve these issues via a carefully designed test function and by applying the maximum principle to the maximal eigenvalue of the Hessian matrix. As applications, we solve two open problems in dimensions 3 and 4. Finally, we give examples to show that these inequalities cannot be extended to $\mathcal{C}_{k-2}$.

math.DG↗

arXiv2026

Uniqueness of capillary Gauss solitons

We prove the rigidity conjecture of [16, Conjecture 1.2] for smooth strictly convex capillary Gauss solitons in a Euclidean half-space with an acute contact angle: every such soliton is a spherical cap. Combined with our previous convergence result for the capillary Gauss curvature flow [16, Theorem 1.1], it follows that the flow starting from a strictly convex capillary hypersurface with an acute contact angle converges to a capillary spherical cap, after a suitable rescaling.

math.DG↗

arXiv2026

The sharp curl-Sobolev inequality

We solve a longstanding problem, going back at least to Rivi\`ere 1998 and open even in the physically most relevant case $n=3$, by proving a sharp curl-Sobolev inequality on $\mathbb{S}^n$ when $n\equiv 3\pmod 4$: for every $\frac{n-1}{2}$-form $\alpha$, the conformally invariant quotient satisfies (with positive denominator) \[ \frac{\Big(\int_{\mathbb{S}^n}|{\rm curl}\alpha|^{\frac{2n}{n+1}}\,{\rm dV}\Big)^{\frac{n+1}{n}}}{\int_{\mathbb{S}^n}\langle{\rm curl}\alpha,\alpha\rangle\,{\rm dV}} \ge \frac{n+1}{2}\,\omega_n^{\frac1n}. \] We also classify all extremals in terms of Killing forms. By conformal invariance, the same result holds on $\mathbb{R}^n$. We then give geometric and variational applications that settle several open conjectures in geometry and mathematical physics. First, we show that on $\mathbb{S}^n$ the round metric is the unique optimizer for the conformal invariant $\mu([g_{{\rm st}}])$. Second, we prove that the unique minimizers of the $3$-energy $\int_{\mathbb{S}^3}|{\rm d} u|^3$ in the homotopy class of the Hopf map $\pi:\mathbb{S}^3\to\mathbb{S}^2$ are exactly $\pi\circ\Phi$ with $\Phi\in{\rm Conf}^+(\mathbb{S}^3)$, confirming a conjecture of Rivi\`ere. Third, for the Faddeev-Skyrme energy $\mathcal{FS}_\rho$ on $\mathbb{S}^3$, we establish global minimality of the Hopf map in the full predicted range: for every coupling constant $\rho\le \sqrt{2}$, the unique global minimizers in its homotopy class are precisely $\pi\circ R$ with $R\in\mathrm{SO}(4)$, as expected since Ward 1999. Fourth, in the presence of Dirac zero modes on $\mathbb{S}^3$, we prove the sharp lower bound $\|{\rm curl} A\|_{3/2} \ge 3\omega_3^{\frac 2 3}$ for the magnetic field and characterize equality in terms of Killing spinors; in particular, this yields a sharp criterion for the existence of zero modes and answers a question of Frank-Loss for $n=3$.

math.DG↗

arXiv2026

On the sharp constants in curl-Sobolev inequalities on $\mathbb{S}^n$

Let $n\equiv 3\ (\mathrm{mod}\ 4)$ and set $p=\frac{n-1}{2}$. On an oriented Riemannian $n$-manifold we consider the (middle-degree) curl operator, $\mathrm{curl}:*\mathrm{d}:\Omega^{p}\rightarrow\Omega^{p}$, and the associated conformally invariant Sobolev quotients on $(\mathbb{S}^n,g_{\mathrm{st}})$, \[ J_1(\alpha)=\frac{\big(\int|\mathrm{curl}\alpha|^{\frac{2n}{n+1}}\,\mathrm{dV}\big)^{\frac{n+1}{n}}}{\int\langle\mathrm{curl}\alpha,\alpha\rangle\,\mathrm{dV}}, \qquad J_2(\alpha)=\frac{\big(\int|\mathrm{curl}\alpha|^{\frac{2n}{n+1}}\,\mathrm{dV}\big)^{\frac{n+1}{n}}}{\inf_{\phi}\big(\int|\alpha-\mathrm{d}\phi|^{\frac{2n}{n-1}}\,\mathrm{dV}\big)^{\frac{n-1}{n}}}. \] Killing $p$-forms and their conformal images form a natural family of critical points for both functionals, analogous to the Aubin-Talenti family in the classical Sobolev inequality. We prove a quantitative local stability estimate for $J_1$ around this family, which in particular implies that every such form is a strict local minimizer in the conformally invariant space $W^{1,\frac{2n}{n+1}}$. In contrast, we show that these critical points are unstable for $J_2$ (and for related conformally invariant quotients), yielding a strict upper bound for the sharp constant of the $J_2$ inequality. By conformal invariance, the results on $\mathbb{S}^n$ transfer naturally to $\mathbb{R}^n$.

math.DG↗

arXiv2026

A new boundary mass for asymptotically flat half-manifolds

We introduce a boundary analogue of the Gauss--Bonnet--Chern mass for asymptotically flat half-manifolds with non-compact boundary. We prove that this mass is well defined and establish the corresponding positive mass theorems for graphical and conformally flat graphs. Also we provide a Penrose-type inequality for the mass $\mathfrak{m}_{a,B}(g)$.

math.DG↗

arXiv2026

The $\sigma_k$-Yamabe problem revisited

In this paper we revisit the $\sigma_k$-Yamabe problem on $M^n$, namely, finding a conformal metric with constant $\sigma_k$-scalar curvature. We prove that on a closed manifold $\left(M,\left[g_0\right]\right)$ with positive Yamabe constant $Y_1\left(M,\left[g_0\right]\right)>0$, the $\sigma_2$-Yamabe constant $$ Y_2\left(M,\left[g_0\right]\right):=\inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} $$ is achieved by a conformal metric $g \in\left[g_0\right]$, which in particular solves the $\sigma_2$-Yamabe problem, assuming $Y_2\left(M,\left[g_0\right]\right)>0$. As a consequence, for any $\left(M, g_0\right)$ with $Y_1\left(M,\left[g_0\right]\right)>$ 0 and $Y_2\left(M,\left[g_0\right]\right)>0$ one has $$ \inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}}=\inf _{g \in\left[g_0\right], R_g>0, \sigma_2(g)>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} . $$ We also show that these conclusions can fail if the condition $R_g>0$ is removed.

math.DG↗

arXiv2026

Blow-up phenomena for the constant Q/R-curvature equation

Let $n\ge 25$ be an integer. In this paper, we construct a Riemannian metric $g_{0}$ on $\mathbb{S}^n$, smooth for $n\geq 26$ and of class $C^9$ but not $C^{10}$ for $n=25$, with the property that the set of metrics in the conformal class of $g_{0}$ having positive scalar curvature and positive constant quotient $Q/R$ is non-compact. Equivalently, we construct families of solutions exhibiting blow-up behavior for the following equation \begin{align*} P _{g_{0}}u- \frac{ (n+2 )(n-4 )}{4} u^{ \frac{2}{n-4}} L_{g_{0}}u^{ \frac{n-2}{n-4}} =0, \quad u>0\quad\text{on} \ \mathbb{S}^{n}, \end{align*} where $P _{g_{0}}$ is the Paneitz operator and $ L_{g_{0}}=-\Delta_{g_{0}} +\frac{n-2}{4(n-1 )}R_{g_{0}} $ is the conformal Laplacian of $ g_{0}$.

math.DG↗

arXiv2026

A Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature

In this paper we introduce the following Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature $R$: Find a conformal metric $g$ in a given conformal class $[g_0]$ with \[ Q_g/R_g=const. \] When the dimension $n\ge 5$, we first prove a new Sobolev inequality between the total $Q$-curvature and the total scalar curvature on $\mathbb{S}^n$ ($n\ge 5$), namely \[\frac{\int_{\mathbb{S}^n} Q_g d v_g}{\left(\int_{\mathbb{S}^n} R_g d v_g\right)^{\frac{n-4}{n-2}}} \geq \frac{\int_{\mathbb{S}^n} Q_{g_{\mathbb{S}^n}} d v\left(g_{\mathbb{S}^n}\right)}{\left(\int_{\mathbb{S}^n} R_{g_{\mathbb{S}^n}} d v\left(g_{\mathbb{S}^n}\right)\right)^{\frac{n-4}{n-2}}}\] for any $g$ in the conformal class of the round metric $g_{\mathbb{S}^n}$ with positive scalar curvature, with equality if and only if $g$ is also a metric with constant sectional curvature. With this inequality we introduce a new Yamabe constant $Y_{4,2}(M,[g_0])$ and prove the existence of the above problem provided that $Y_{4,2}(M,[g_0]) <Y_{4,2} (\mathbb{S}^n, [g_{\mathbb{S}^n}]).$ This strict inequality is proved if $(M,g)$ is not conformally equivalent to the round sphere. This follows from a crucial relation between $Y_{4,2}$ and the ordinary Yamabe constant $Y(M,[g_0])$, $Y_{4,2} (M, [g_0]) \le c(n) Y(M, [g_0])^{\frac n{n-2}}$ with equality if and only if $(M, g_0)$ is conformally equivalent to an Einstein manifold. Finally, we prove that on a closed $n$-dimensional Riemannian manifold $(M,g_{0})$ with semi-positive $Q$-curvature and non-negative scalar curvature, the above Yamabe problem is solvable, thanks to the maximum principle of Gursky-Malchiodi [33]. The proof for $n=3$ and $n=4$ follows closely the methods developed by Hang-Yang in [40], Gursky-Malchiodi in [33], and Chang-Yang in [12].

math.DG↗

arXiv2026

A flow approach to the Toda system

In this paper we introduce a flow to study the Toda system, which we call {\it Toda flow.} More generally, we introduce a flow of the Liouville systems, formulated as a coupled parabolic system with nonlocal interactions. Finite-time singularities are characterized and both necessary and sufficient conditions for convergence are provided in this general setting, even when the prescribed functions are allowed to change sign. As an application, we prove a global existence for the Toda flow in the critical case without restricting the sign of the prescribed functions. We provide a detailed description of blow-up behavior at infinity and obtain a sharp lower bound for the functional in cases where global convergence fails. By constructing appropriate test functions, we further establish a sufficient condition for the global convergence of the flow. These results are not affected by the sign-changing nature of the prescribed functions, and extend the theorem of Jost, Lin and Wang (Comm. Pure Appl. Math. 59, 526-558, 2006) to systems of multiple equations under this more general and physically relevant condition.

math.DG↗

arXiv2026

A half-space Liouville theorem for anisotropic minimal graph with free boundary

In this paper we prove the following Liouville-type theorem: any anisotropic minimal graph with free boundary in the half-space must be flat, provided that the graph function has at most one-sided linear growth. This extends the classical results of Bombieri-De Giorgi-Miranda and Simon to an appropriate free boundary setting.

math.DG↗

arXiv2025

Conformal invariants for the zero mode equation

For non-trivial solutions to the zero mode equation on a closed spin manifold \[D \varphi=iA\cdot \varphi,\] we first provide a simple proof for the sharp inequality \eq{ \norm{A}_{L^n}^2 \ge \frac {n}{4(n-1)} Y(M,[g]), } where $Y(M,[g])$ is the Yamabe constant of $(M,g)$, which was obtained by Frank-Loss and Reuss. Then we classify completely the equality case by proving that equality holds if and only if $\varphi$ is a Killing spinor, and if and only if $(M,g)$ is a Sasaki-Einstein manifold with $A$ (up to scaling) as its Reeb field and $\varphi$ a vacuum up to a conformal transformation. More generalizations have been also studied.

math.DG↗

arXiv2025

Prescribed $L_{p}$ curvature problem for convex capillary hypersurface

We address the prescribed $L_p$ curvature problem for convex capillary hypersurfaces in the Euclidean half-space. By reducing it to a convex solution of a Hessian quotient equation on a spherical cap with a Robin boundary condition, we establish the existence and uniqueness of smooth admissible, and indeed strictly convex, solutions. In particular, we solve the capillary $L_p$ Christoffel--Minkowski problem for $p\geq 1$ in the smooth category, providing a natural Robin boundary counterpart of the classical $L_p$ Christoffel--Minkowski problem of Hu--Ma--Shen [25] and Guan--Xia [24]. We further obtain analogous existence and uniqueness results for the prescribed $L_p$ curvature problem and the associated eigenvalue problem for convex capillary hypersurfaces in the Euclidean half-space. These results form a capstone to our series of works [48,49,51].

math.DG↗

arXiv2025

The capillary Christoffel-Minkowski problem

In this article, we introduce a $k$-th capillary area measure for capillary convex bodies in the Euclidean half-space, which serves as a boundary counterpart to the classical concept of area measure (see, e.g., \cite[Chapter 8]{Sch}). We then propose a Christoffel-Minkowski problem for capillary convex bodies, to find a capillary convex body in the Euclidean half-space with a prescribed $k$-th capillary area measure. This problem is equivalent to solving a Hessian-type equation with a Robin boundary value condition. We then establish the existence and uniqueness of a smooth solution under a natural sufficient condition.

math.AP↗

arXiv2025

Stability of spinorial Sobolev inequalities on $\mathbb{S}^n$

The spinorial Sobolev inequality on the unit sphere states \begin{equation*} \Big(\int| D\psi|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}-\frac{n}{2}\omega_{n}^{1/n}\int\langle D\psi,\psi\rangle \geq 0, \end{equation*} with equality if and only if $\psi \in {\mathcal M}$, the set of all $-\frac 12$-Killing spinors and their conformal transformations. Our main result in this paper is to refine this inequality by establishing a stability inequality \begin{equation*} \Big(\int| D\psi|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}-\frac{n}{2}\omega_{n}^{1/n}\int\langle D\psi,\psi\rangle \geq {\bf c}_S\inf_{\phi\in\mathcal{M}}\Big(\int| D(\psi-\phi)|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}. \end{equation*} As a by-product of our argument, we show that elements in set $\mathcal M$ are not optimizers of another spinorial Sobolev inequality \begin{equation*} \Big(\int| D\psi|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}} \geq C_S \Big(\int|\psi|^{\frac{2n}{n-1}}\Big)^{\frac{n-1}{n}}, \end{equation*} unlike expected by experts. They have in fact index $n+1$ and nullity $2^{[\frac n2]+2}$.

math.DG↗

arXiv2025

The capillary Gauss curvature flow

In this article, we first introduce a Gauss curvature type flow for capillary hypersurfaces, which we call capillary Gauss curvature flow. We then show that the flow will shrink to a point in finite time. This is a capillary counterpart (or Robin boundary counterpart) of Firey's problem studied in [Mathematika 21 (1974), pp. 1-11] and Tso [Comm. Pure Appl. Math. 38 (1985), no. 6, 867-882]. Finally, we prove that its normalized flow converges to a soliton. This is a capillary counterpart of the result of Guan and Ni in [J. Eur. Math. Soc. 19 (2017), no. 12, 3735-3761]. The classification of solitons remains an open conjecture.

math.DG↗

arXiv2025

Half-space Liouville-type theorems for minimal graphs with capillary boundary

In this paper, we prove two Liouville-type theorems for capillary minimal graph over $\mathbb{R}^n_+$. First, if $u$ has linear growth, then for $n=2,3$ and for any $\theta\in(0,\pi)$, or $n\geq4$ and $\theta\in(\frac{\pi}6,\frac{5\pi}6)$, $u$ must be flat. Second, if $u$ is one-sided bounded on $\mathbb{R}^n_+$, then for any $n$ and $\theta\in(0,\pi)$, $u$ must be flat. The proofs build upon gradient estimates for the mean curvature equation over $\mathbb{R}^n_+$ with capillary boundary condition, which are based on carefully adapting the maximum principle to the capillary setting.

math.DG↗

arXiv2025

The capillary $L_p$-Minkowski problem

This paper is a continuation of our recent work [Adv. Math. 469 (2025), Paper No. 110230] concerning the capillary Minkowski problem. We propose, in this paper, a capillary $L_p$-Minkowski problem for $p\in \mathbb{R}$, which seeks to find a capillary convex body with a prescribed capillary $L_p$-surface area measure in the Euclidean half-space. This formulation provides a natural Robin boundary analogue of the classical $L_p$-Minkowski problem introduced by Lutwak [J. Differential Geom. 38 (1993), no. 1, 131--150]. For $p>1$, we resolve the capillary $L_p$-Minkowski problem in the smooth category by reducing it to a Monge--Amp\`ere equation with a Robin boundary condition on the unit spherical cap.

math.DG↗

arXiv2025

Convex capillary hypersurfaces of prescribed curvature problem

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].

math.DG↗