微分几何

We describe the Poisson geometry of the Madelung transform between quantum mechanics and hydrodynamics for generic wave functions. We prove that for arbitrary oriented manifolds this transform, being regarded as a momentum map, naturally…

We reformulate binary classification on a manifold M as a Yang-Mills-Higgs variational problem. Labelled data is encoded as a functor from the fundamental groupoid of M to the one-object groupoid B(Z_2), whose monodromy class in H^1(M, Z_2)…

We propose a dictionary between binary classification in machine learning and differential geometry. Classifiers are parallel sections of vector bundles over the data space; training labels become Dirichlet boundary conditions; the kernel…

We prove a sharp Veronese rigidity theorem for closed immersed submanifolds of the Euclidean unit ball under intrinsic harmonic-structure assumptions. For an isometric immersion $F:(\Sigma,g)\looparrowright\overline B(1)$, define the…

In this paper, we classify all solitons of the Gauss curvature flow in the three-dimensional Heisenberg group $\mathrm{Nil}_3$ that are invariant under a one-parameter group of ambient isometries. By means of the four canonical types of…

We study a notion of fractional $s$-mass for codimension-two currents on closed Riemannian manifolds, defined via energy minimization with a prescribed Jacobian constraint. We prove equi-coercivity and $\Gamma$-convergence, with respect to…

We study constant mean curvature hypersurfaces in $\mathbb{H}^2\times\mathbb{H}^2$ invariant under a double horocyclic action. We show that the CMC condition reduces to a single autonomous ordinary differential equation for an angular…

We study oriented surfaces in the Heisenberg space $\mathrm{Nil}_3$ whose mean curvature $H$ at each point is $H=\langle N,\partial_z\rangle+\lambda$, where $N$ is the unit normal, $\partial_z$ is the vertical Killing vector field and…

A $\lambda$-translator in $\mathbb{S}^2\times\mathbb{R}$ is an oriented surface whose mean curvature $H$ satisfies $H=\langle N,\partial_z\rangle+\lambda$, where $N$ is the unit normal, $\partial_z$ is the vertical Killing vector field…

This paper establishes Bismut-type formulas and gradient estimates for Feynman--Kac semigroups on Riemannian manifolds with boundary, under geometric conditions formulated in terms of Ricci curvature $\mathrm{Ric}_Z \geq K$ and second…

In this note we prove a conjecture by Hasegawa stating that a simply connected, nilpotent Lie group of dimension $2n$ endowed with a left-invariant complex structure is biholomorphic to $\mathbb{C}^n$.

We relate the hyperbolicity of a calibrated manifold $(X, \phi)$ to the analytic properties of the space of Smith immersions $\mathrm{SmIm}(B^k, X)$ from the Poincare $k$-ball into $X$. In particular, we establish the following calibrated…

We develop an analytic construction of nowhere-vanishing harmonic $1$-forms on real loci of K3-fibred Calabi-Yau $3$-folds with collapsing Ricci-flat K\"ahler metrics. We apply our construction to examples whose real loci have connected…

Motivated by the recent work of Lipton on the M\"obius energy of helix curves, we extend the study to the $L^p$ asymptotics of the meromorphic family \[ M_\rho(t) = \frac{\rho^2+1}{\rho^2 t^2 + 4 \sin^2(t/2)} - \frac{1}{t^2}. \] The helix…

We construct infinitely many embedded minimal hypersurfaces of pairwise distinct irreducible topological types in the unit $4$-sphere $\mathbb{S}^4$, which provides a new answer to a problem of Hsiang. These examples are topologically…

We investigate four-dimensional electrostatic systems arising as spatial factors of static Einstein--Maxwell spacetimes with cosmological constant. Assuming that the electric field is everywhere collinear with the gradient of the lapse…

We extend the Sternberg--Weinstein coupling construction to the Jacobi geometry setting. Starting from a Jacobi Hamiltonian $G$-space and a principal bundle equipped with a connection whose curvature satisfies some nondegeneracy condition,…

Lu's first pinching theorem states that a closed minimal $n$-dimensional submanifold of the unit sphere satisfying $0\le S+\lambda_2\le n$ is one of the standard first-gap models; here $S$ is the squared norm of the second fundamental form…

We present the entropy-degree theorem for Lipschitz maps between Alexandrov spaces with curvature bounded below, extending the classical Besson--Courtois--Gallot entropy-rigidity results to this singular setting. The proof requires a new…

We compute invariants of Carrollian spacetimes, deriving them from the geometry of the screen bundle. For generic Carrollian structures we specify how to generate the entire algebra of differential invariants, with emphasis on dimension 3,…