微分几何

Hilbert's fourth problem seeks the classification of metric geometries where straight lines are shortest paths. Its regular case identifies the projectively flat Finsler manifolds. This broader framework breaks the equivalence between…

We prove that the four-dimensional blowdown shrinking Ricci soliton constructed by Feldman-Ilmanen-Knopf is strictly linearly stable in the sense of Cao-Hamilton-Ilmanen. This provides the first known example of a non-cylindrical linearly…

We provide various characterizations for a given almost smooth space to be an RCD space, in terms of a local volume doubling and a local Poincar\'e inequality. Applications include a characterization of Einstein $4$-orbifolds.

In this article, we study geometric and analytical features of complete noncompact $\rho$-Einstein solitons, which are self-similar solutions of the Ricci-Bourguignon flow. We study the spectrum of the drifted Laplacian operator for…

We introduce the space of mixed-volume forms endowed with a $L^2$ metric on a balanced manifold. A geodesic equation can be derived in this space that has an interesting structure and extends the equation of Donaldson \cite{Donaldson10} and…

We study the regularized determinants ${\rm det}\, \Delta$ of various self-adjoint extensions of symmetric Laplacians acting in spinor bundles over compact Riemann surfaces with flat singular metrics $|\omega|^2$, where $\omega$ is a…

While $L_\infty$ algebras are fundamental structures in differential geometry and mathematical physics, the geometric information encoded in such structures is often implicit. We address the following question: What constitutes a…

We explain a construction of $G_2$-instantons on manifolds obtained by resolving $G_2$-orbifolds. This includes the case of $G_2$-instantons on resolutions of $T^7/\Gamma$ as a special case. The ingredients needed are a $G_2$-instanton on…

In this article, we examine continuity properties of the maps $|Hol$ and $\Hol^0$ assigning, on a fixed manifold $M$, to a metric on $M$ its holonomy class resp. restricted holonomy class (conjugacy class of the connected component of the…

This paper provides a classification of analytic actions of the semi-orthogonal group $\text{SO}^\circ(p,q)$, for $p,q \geq 3$, on closed, connected $(p+q-1)$-dimensional manifolds. Adapting Uchida's construction of $\text{SO}^\circ(p,q)$…

We study the gradient flow of Spin($7$)-structures and construct the first explicit solutions, in the homogeneous setting. As an intermediate step, we obtain formulae expressing the Spin($7$)-torsion tensor and gradient flow in terms of the…

This is a sequel to a series of works, where we studied the local aspects of the asymptotic action of deformation quantization on the Hilbert spaces $H^0(X, L^{\otimes k})$ of geometric quantization for a K\"ahler manifold $X$; here $L$ is…

We give a sufficient criterion, which we call stability, for a coarse Lipschitz map $f$ from a complete manifold $X$ with Ricci curvature bounded below to a proper Hadamard space $Y$ to be within bounded distance of a harmonic map. We prove…

We establish a duality between harmonic maps from Riemann surfaces to hyperbolic 3-space $\mathbb{H}^3$ and harmonic maps from Riemann surfaces to de Sitter three-space $\operatorname{dS}_3$, best viewed as a generalized Gauss map. On the…

We develop the theory of $H$-graded manifolds for any finitely generated abelian group, using tools from representation theory. Furthermore, we introduce and investigate the notion of $H$-graded coverings of supermanifolds in the case where…

To classify a group action on a manifold, the data associated with the fixed point set is essential. In this paper, we classify the fixed point data of a circle action on a 6-dimensional compact connected oriented manifold with isolated…

We develop an intrinsic information-theoretic approach for recovering Riemannian curvature from the small-time behaviour of heat diffusion. Given a point and a two-plane in the tangent space, we compare the heat mass transported along that…

The elastic properties of a material are encoded in a stiffness tensor field and the propagation of elastic waves is modeled by the elastic wave equation. We characterize analytic and algebraic properties a general anisotropic stiffness…

In this study, the geometric properties of null helices on a totally umbilical submanifold within a three-dimensional semi-Riemannian manifold are investigated. The pseudo-Riemannian metric structure of semi-Riemannian manifolds and the…

We consider the pseudo-Riemannian Lichnerowicz conjecture in the homogeneous setting. In particular, we show that any compact connected pseudo-Riemannian manifold $M$ on which a semisimple group $G$ acts conformally, essentially and…