微分几何

Many geometric machine learning and image analysis applications, require a left-invariant metric on the 5D homogeneous space of 3D positions and orientations SE(3)/SO(2). This is done in Equivariant Neural Networks (G-CNNs), or in PDE-Based…

We propose a construction of mean curvature flows by approximation for very general initial data, in the spirit of the works of Brakke and of Kim & Tonegawa based on the theory of varifolds. Given a general varifold, we construct by…

In this paper, we extend the notion of stationary curves with respect to the moment of inertia from a point $N$ in the Euclidean plane $\mathbb{R}^2$ to the case that the ambient space is either the hyperbolic plane $\mathbb{H}^2$ or the…

Under natural assumptions on curvature and cross section, we establish the uniqueness of asymptotic limits and the exponential convergence rate for complete noncollapsed Ricci-flat manifolds with linear volume growth, which are known to…

In this paper, the directional derivatives in accordance with the orthonormal frame {T, N, B} are defined in $M_{q}^{3}(c)$, and the extended Serret-Frenet relations by using Frenet formulas are expressed. Furthermore, we express the…

Given any K\"ahler manifold $X$, Kapranov discovered an $L_\infty[1]$ algebra structure on $\Omega^{0,\bullet}_X(T^{1,0}_X)$. Motivated by this result, we introduce, as a generalization of $L_\infty[1]$ algebras, a notion of $L_\infty[1]$…

A hypersurface $M$ in the unit sphere $S^n \subset {\bf R}^{n+1}$ is Dupin if along each curvature surface of $M$, the corresponding principal curvature is constant. If the number $g$ of distinct principal curvatures is constant on $M$,…

Let $X$ be a smooth complex manifold. Assume that $Y\subset X$ is a K\"{a}hler submanifold such that $X\setminus Y$ is biholomorphic to $\mathbb{C}^n$. We prove that $(X, Y)$ is biholomorphic to the standard example $(\mathbb{P}^n,…

We construct an embedded ancient solution to the mean curvature flow which is qualitatively given by a ``stack'' of two ancient pancakes joined by a neck. Our solution is closed, non-convex, and contained in a slab.

Fermat's principle is fully generalized to the case where a smooth interface separates two cone structures -- Lorentz-Finsler lightcones -- representing wave propagation in a potentially inhomogeneous, anisotropic, time-dependent and…

We adopt a measure-theoretic perspective on the Riemannian approximation scheme proving a sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds. We show that the zero-order term in the limit is a singular measure…

In this work we provide a complete characterization of left-invariant symmetric Killing tensors on almost abelian Lie groups endowed with a left-invariant Riemannian metric. We show in particular that all such tensors are decomposable, in…

On the space of isometric embeddings $f_g$ of metrics $g$ on a manifold $M^n$ into the standard $(\mb{S}^{\tn=\tn(n)},\tg)$, we consider the total exterior scalar curvature $\Theta_{f_g}(M)$, and squared $L^2$ norm of the mean curvature…

We study a CR-invariant equation for vanishing $E_1$ surfaces in the 3-dimensional Heisenberg group. This is shown to be a hyperbolic equation. We prove the local uniqueness theorem for an initial value problem and classify all such global…

Consider $\mathbb{R}^3$ equipped with the Euclidean metric and the Gaussian measure. Let $\Sigma$ be a complete embedded self-shrinker in $\mathbb{R}^3$ with the induced metric and weighted measure, and let $\lambda_1$ denote the first…

Chekanov's exotic tori have been playing an important role in symplectic geometry as they are the only known examples of Lagrangian tori in ${\mathbb{C}}^2$ that are not Hamiltonian isotopic to a product torus. In this paper, we explore the…

In the infinite-dimensional Banach setting, we consider general smooth Banach fibrations $\tau:M\to M_0$ and `$(1,1)$-tensors' $N:TM\to TM$ that are projectable (in the obvious sense) onto Nijenhuis operators $N_0:TM_0\to TM_0$ on $M_0$. We…

We prove that the combination of strict positivity of $k$-tri-Ricci curvature with non-negative $3$-intermediate Ricci curvature forces rigidity of two-sided stable free boundary minimal hypersurface in a 5-manifold with bounded geometry…

We provide some examples of Killing superalgebras on 2-dimensional pseudo-Riemannian manifolds within the theoretical framework established in [SIGMA 21 (2025), 081, 61 pages, arXiv:2409.11306]. We compute the Spencer cohomology group…

We generalise the notion of a Killing superalgebra, which arises in the physics literature on supergravity, to general dimension, signature and choice of spinor module and Dirac current. We also allow for Lie algebras as well as…