微分几何

In this paper we prove that a metric measure space $(X,d,m)$ satisfying the finite Riemannian curvature-dimension condition ${\sf RCD}(K,N)$ is non-branching and that tangent cones from the same sequence of rescalings are H\"older…

The Eguchi-Hanson metric is a natural metric on the total space of the cotangent bundle $T^*\mathbb{CP}(1)$ of the complex projective line $\mathbb{CP}(1) \simeq \mathbb{S}^2$, which extends the Fubini-Study metric of $\mathbb{CP}(1)$. By…

In this article we prove that the set of flat singular points of locally highest density of area-minimizing integral currents of dimension $m$ and general codimension in a smooth Riemannian manifold $\Sigma$ has locally finite…

In this paper, we introduce a new canonical connection on Riemannian manifold with a distribution. Moreover, as an application of the connection, we give a geometric proof of the Frobenius theorem.

Let $M$ be a compact, orientable, $n$-dimensional Riemannian manifold, $n\geq2$, and let $F$ be the energy functional acting on the space $\Xi (M)$ of $C^{\infty }$ vector fields of $M$, \[ F(X):=\frac{\int_{M}\left\Vert \nabla X\right\Vert…

In 1961, Palais showed that every smooth proper Lie group action on a smooth manifold admits a compatible Riemannian metric on the manifold such that the action becomes isometric. In 2006, Yoshino studied a continuous proper action of a…

We give conserved quantities of two generalizations of conformal circles on the conformal sphere. One generalization concerns curves carrying a distinguished parallel tractor along them, which can be used to construct the conserved…

We show that, up to biholomorphism, a given noncompact complex manifold only admits one shrinking gradient K\"ahler-Ricci soliton with Ricci curvature tending to zero at infinity. Our result does not require fixing the asymptotic data of…

Motivated by applications to holography and Teichm\"uller theory, we prove a continuous analogue of the max flow/min cut theorem which also takes the topology of the domain into account.

In this paper, we firstly provide a concise overview of $\mathcal{S}-$manifolds, $f$-biharmonicity and $\theta _{\alpha }$-slant curves. We then derive a key equation and analyze it in detail to establish the necessary and sufficient…

We consider general integrable curve nets in Euclidean space as a particular integrable geometry invariant with respect to rigid motions and net-preserving reparameterisations. For the purpose of their description, we first give an overview…

We compute the Laplacian spectra of singular area-minimising hypersurfaces in the hyperbolic space with prescribed asymptotic data. We also obtain similar results in higher codimension, and explore related extremal properties of the bottom…

Let $X$ be a compact K\"ahler manifold and $D$ be a simple normal crossing divisor on $X$ such that $K_X+D$ is big and nef. We first prove that the singular K\"ahler--Einstein metric constructed by Berman--Guenancia is almost-complete on $X…

Let $G$ be a connected, linear real reductive group and let $X$ be a cocompact $G$-proper manifold without boundary. We define delocalized eta invariants associated to a $L^2$-invertible perturbed Dirac operator $D_X+A$ with $A$ a suitable…

These are the notes for a series of lectures at the Institute of Geometry and Topology of the University of Stuttgart, Germany, in July 13-15, 2022. We assume basic knowledge of isometric actions on Riemannian manifolds, including the…

We consider the Rumin complex associated with a generic rank two distribution on a closed 5-manifold. The Rumin differential in middle degrees gives rise to a self-adjoint differential operator of Heisenberg order two. We study the eta…

We prove a parabolically scale-invariant variation of the planarity estimate in \cite{Na22} for higher codimension mean curvature flow, borrowing ideas from work of Brendle--Huisken--Sinestrari \cite{BHS}. Additionally, we prove convexity…

In this paper, we investigate the problem of $\mathcal{F}$-harmonic forms and the long-time behavior of the Type IIA flow on certain nilpotent and solvable symplectic Lie algebras (and corresponding nilmanifolds and solvmanifolds). In…

In the framework of sub-Riemannian Manifolds, a relevant question is: what are the \enquote{metric lines} (i.e., the isometric embedding of the real line)? This article presents a conjecture classifying the metric lines in Carnot groups and…

The Kerr-star spacetime is the extension over the horizons and in the negative radial region of the slowly rotating Kerr black hole. It is known that below the inner horizon, there exist both timelike and null (lightlike) closed curves.…