微分几何

In this paper, we establish the existence of prescribed mean curvature (PMC) hypersurfaces in conformal product manifolds with (possibly empty) $C^{1,\alpha}$ fixed graphical boundaries under a barrier condition. This result generalizes…

A singular foliation $\mathcal{F}$ on a complete Riemannian manifold $M$ is called Singular Riemannian foliation (SRF for short) if its leaves are locally equidistant, e.g., the partition of $M$ into the orbits of a Lie group action by…

We study Lie bialgebra structures on \emph{flat metric Lie algebras}, that is, Lie algebras $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ whose associated left-invariant Riemannian metric on the simply connected Lie group $G$ has zero…

A pair of tensors $(g,B)$ form the induced metric and shape operator of an immersion into hyperbolic space if and only if they satisfy the Gauss-Codazzi equations. Such a pair of tensors induce a pair $(\hat{g},\hat{B})$ related to the…

In this paper we establish a positive lower bound estimate for the second smallest eigenvalue of the complex Hessian of solutions to a degenerate complex Monge-Amp\`ere equation. As a consequence, we find that in the space of K\"ahler…

For the largest exceptional simple Lie superalgebra $F(4)$, having dimension $(24|16)$, we provide two explicit geometric realizations as supersymmetries, namely as the symmetry superalgebra of super-PDE systems of second and third order…

We investigate the relationship between two interpretations of equivariant Riemann-Roch defects of complex spaces with conic singularities; as (i) equivariant $\eta_{T}$ and $\xi_{T}$ invariants, and as (ii) supertraces over local…

We prove uniqueness results for capillary disks in three-dimensional domains that are modeled by an elliptic PDE, under the assumption that the domain admits a family of surfaces with suitable properties. Our main theorem generalizes…

In this paper, we prove the Riemannian positive mass theorem up to dimension $19$, building on a combination of torical symmetrization and the singularity blow-up technique developed in [HSY26], together with the generic regularity theory…

On an almost complex manifold $(M,J)$, a pluricanonical locally conformally almost K\"ahler (LCAK) metric $g$ is induced by a locally conformally symplectic structure $(F,\theta)$ of the first kind, characterized by the fact that $D\theta$…

In this series of two papers we will generalise the concept of extending a Lie algebroid by a Lie algebra bundle, leading to a notion of extending a Lie algebroid by another Lie algebroid whose orbits lie in the orbits of the former…

This note corrects an error in the proof of Proposition 13 in arXiv:1903.09181 and simultaneously establishes a more general result. We prove that if $M $ is a compact connected oriented $4$-manifold with connected boundary $\partial M$,…

Due to its rich structure and close connection with gauge theory, hyperk\"ahler manifolds have attracted increasing interest. Using infinite dimensional hyperk\"ahler reduction, Kronheimer proved that certain adjoint orbits of complexified…

We study the stability and H\"older continuity of solutions to degenerate complex Monge--Amp\`ere equations associated with a (non-closed) big form on compact Hermitian manifolds. We also show that the solution is globally continuous when…

For the de Rham mapping cone cochain complex induced by a smooth closed 2-form, we explicitly write down the associated mapping cone Thom form in the sense of Mathai-Quillen. Our construction uses the mapping cone covariant derivative,…

We construct an example of an asymptotically conical (AC) non-K\"ahler expanding gradient Ricci soliton that has a K\"ahler tangent cone at infinity. This yields an example of a K\"ahler cone that can be desingularised by a smooth AC…

Using discretized orthogonal systems (curvature line systems) with periodicity, created using Darboux transformations and their permutability, we have discrete and semi-discrete k-dimensional isothermic tori which are full in n-dimensional…

We study zero-divisors in the $16$-dimensional sedenion algebra from the viewpoint of the determinant of left multiplication. We show that this determinant admits a canonical factorization into the square of a quartic polynomial, obtained…

We review recent progress concerning the analysis of Lagrangians on immersions into $\mathbb{R}^d$ depending on the first and second fundamental forms and their covariant derivatives.

A Weyl structure on a Riemannian manifold $(M,g)$ is a torsion-free linear connection $\nabla$ such that there is a $1$-form $\theta$ (called the Lee form) satisfying $\nabla g = 2\, \theta \otimes g$. We examine the case in which there…