微分几何

In work with P. Chru\'sciel, L. Nguyen and T.-T. Paetz [8], a positive mass theorem was obtained for asymptotically locally hyperbolic manifolds with boundary, having a toroidal end. The proof made use of properties of marginally outer…

For a compact Riemannian surface $(M,g)$ with non-empty boundary $\Gamma$, the Dirichlet-to-Neumann operator (DtN-map) $\Lambda_g:C^\infty(\Gamma)\to C^\infty(\Gamma)$ is defined by $\Lambda_gf=\left.\frac{\partial…

We determine all four-dimensional Lie groups which have harmonic curvature. As a consequence, a description of four-dimensional pp-wave Lie groups is obtained.

The conformal-to-Einstein operator is a conformally invariant linear overdetermined differential operator whose non-vanishing solutions correspond to Einstein metrics within a conformal class. We construct compatibility complexes for this…

In this paper, we investigate uniform spectral gaps for Weil-Petersson random hyperbolic surfaces with not many cusps. We show that if $n=O(g^\alpha)$ where $\alpha\in \left[0,\frac{1}{2}\right)$, then for any $\epsilon>0$, a random cusped…

We obtain generalized Wintgen inequalities for submanifolds in conformally flat manifolds. We give some applications for submanifolds in a Riemannian manifold of quasi-constant curvature. Equality cases are also considered.

The full asymptotic expansion of the equivariant complex Ray-Singer torsion for high powers of line bundles on symmetric spaces is given in an explicit form. In the case of isolated fixed points this expansion is given for general complex…

This paper investigates the isomorphisms between principal distributions $\mathcal{D}_k$ $(k=1,\dots 4)$ on OT--FKM type isoparametric hypersurfaces in spheres. We recover the isomorphism $\mathcal{D}_1 \cong \mathcal{D}_3$ established by…

Wang, Weng and Xia[Math. Ann. 388 (2024), no. 2] studied a mean curvature type flow for the smooth, embedded capillary hypersurfaces with a constant contact angle $\theta\in(0,\pi)$ and confirmed the existence of solutions by the standard…

In this paper, we establish some comparison theorems for the total quotient curvature. Specifically, we examine the behavior of the functional with respect to the total quotient curvature and prove that the background Einstein metric…

Motivated by Osserman's problem on the number $D_g$ of omitted values of the Gauss map of a complete minimal surface with finite total curvature in $\boldsymbol{R}^3$, its totally ramified value number $\nu_g$ (referred to in this paper as…

We prove that a compact Riemannian manifold of dimension $n\ge 8$ with harmonic Weyl curvature and $\frac{3(n-1)(n+2)}{4(3n-1)}$-nonnegative curvature operator of the second kind is either globally conformally equivalent to a space of…

In the first part of the paper, we prove the existence of longtime solution to mean curvature flow starting from a graph of a continuous function defined over a slab. Then, we establish dynamical stability results for various types of…

In a series of papers, Weil initiated the investigation of translation invariant curvature measures of convex bodies, which include as prime examples Federer's curvature measures. In this paper, we continue this line of research by…

Gromov conjectured that the total mean curvature of the boundary of a compact Riemannian manifold can be estimated from above by a constant depending only on the boundary metric and on a lower bound for the scalar curvature of the fill-in.…

We prove that complete Riemannian manifolds of dimension $n\ge3$ with harmonic curvature and $\frac{n(n+2)}{2(n+1)}$-nonnegative curvature operator of the second kind must be Einstein. In particular, We show that complete Einstein manifolds…

We study rotationally symmetric translators for fully nonlinear extrinsic geometric flows driven by a curvature function, and we establish the fine asymptotics of bowl-type evolutions and, when admissible, the construction and…

Using Bochner techniques, we prove that a compact Einstein manifold of dimension $n \ge 4$ has constant curvature provided that the curvature operator of the second kind satisfies a cone condition that is strictly weaker than nonnegativity.…

We prove that a vertical plane is the only complete translator, properly immersed in $\mathbb{R}^3$ and having finite topology, whose ends are asymptotic to vertical planes.

Motivated by Schoen's conjecture on the volume functional for closed hyperbolic manifolds, we generalize the volume comparison theorem of Hu, Ji, and Shi and establish a volume comparison theorem for rank 1 symmetric spaces of non-compact…