arXiv subjects

Jinyu Gao

Publications and source records attributed to Jinyu Gao.

4 recordsLinked to original sources

arXiv2026

The Anisotropic Capillary $L_p$-Minkowski Problem

This paper introduces the \textit{anisotropic $\omega_0$-capillary $p$-sum} of two hypersurfaces in $\mathbb{R}_+^{n+1}$, and establishes a theory for anisotropic capillary convex bodies. For a smooth convex hypersurface $\Sigma $ with anisotropic $\omega_0$-capillary boundary, we compute the variation of its anisotropic capillary $k$-th quermassintegral via this $p$-sum, thereby defining the associated anisotropic $\omega_0$-capillary $k$-th $p$-surface area measure on the capillary Wulff shape $\mathcal{C}_{\omega_{0}}$. This motivates us to propose and solve the anisotropic capillary $L_{p}$-Minkowski problem for $p\geq1$.

math.DG↗

arXiv2025

Alexandrov-Fenchel inequalities for convex anisotropic capillary hypersurfaces in the half-space

In this paper, the results of Mei, Wang, Weng and Xia [Math. Z., 2025, MR4911815] on capillary convex bodies are extended to the anisotropic setting. We develop a theory for anisotropic capillary convex bodies in the half-space and establish a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies. Thus, this weakens the conditions of the inequality in [Ding-Gao-Li,arXiv:2408.10740, Theorem 1.4] and extends it to a more general case.

math.DG↗

arXiv2025

Generalized Minkowski formulas and rigidity results for anisotropic capillary hypersurfaces

In this paper, we obtain a new Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces in the half-space, which includes the weighted Hsiung-Minkowski formula and classical anisotropic Minkowski identity for closed hypersurfaces as special cases. As applications, we prove some anisotropic Alexandrov-type theorems and rigidity results for anisotropic capillary hypersurfaces. Specially, the uniqueness of the solution to the anisotropic Orlicz-Christoffel-Minkowski problem is obtained, and thus a new proof is provided for the uniqueness of the solution to $L_p$-Minkowski problem with $p\geq 1$ in the Euclidean capillary convex bodies geometry.

math.DG↗

arXiv2025

Anisotropic mean curvature type flow and capillary Alexandrov-Fenchel inequalities

In this paper, an anisotropic volume-preserving mean curvature type flow for star-shaped anisotropic $ω_0$-capillary hypersurfaces in the half-space is studied, and the long-time existence and smooth convergence to a capillary Wulff shape are obtained. If the initial hypersurface is strictly convex, the solution of this flow remains to be strictly convex for all $t>0$ by adopting a new approach applicable to anisotropic capillary setting. In analogy with closed hypersurfaces, if the $ω_0$-capillary Wulff shape is a $θ$-capillary hypersurface with constant contact angle $θ$, the quermassintegrals for anisotropic capillary hypersurfaces match the mixed volume of two $θ$-capillary convex bodies. Thus, generalized quermassintegrals for anisotropic capillary hypersurfaces with general Wulff shapes (i.e., the $ω_0$-capillary Wulff shape has a variable contact angle) can be defined, which satisfy certain monotonicity properties along the flow. As applications, we establish an anisotropic capillary isoperimetric inequality for star-shaped anisotropic capillary hypersurfaces and a family of new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces. In particular, we provide a flow's method to derive the Alexandrov-Fenchel inequalities for two $θ$-capillary hypersurfaces, demonstrated in [30] (arXiv:2408.13655) from the view of point in convex geometry.

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