微分几何

We obtain a generic regularity result for stationary integral $n$-varifolds with only strongly isolated singularities inside $N$-dimensional Riemannian manifolds, in absence of any restriction on the dimension ($n\geq 2$) and codimension.…

We show that there exists a quantity, depending only on $C^0$ data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian…

We construct a countable collection of one-parameter families of non-rotational minimal annuli with free boundary in geodesic balls of hyperbolic 3-space. Every surface within a given family shares a common prismatic symmetry group, and…

In this paper, we construct a stable parabolic Higgs bundle of rank two, which corresponds to the uniformization associated with a conformal hyperbolic metric on a compact Riemann surface $\overline{X}$ with prescribed singularities. This…

Let f = 0 be an implicit singular plane curve. When deforming f = 0, inflections and vertex emerge from the singularities. In this papper, we classify the deformations of f = 0 with respect to the inflections and the vertices in the cases…

Let $(\Omega^{n+1},g)$ be an $(n + 1)$-dimensional smooth complete connected Riemannian manifold with compact boundary $\partial\Omega=\Sigma$ and $f$ a smooth function on $\Omega$ which satisfies the Obata type equation $\nabla^2 f -fg =0$…

We consider balanced metrics on complex manifolds with holomorphically trivial canonical bundle, most commonly known as balanced $\rm{SU}(n)$-structures. Such structures are of interest for both Hermitian geometry and string theory, since…

This work studies simply connected, noncompact $G/H$ in which $G$ is semi-simple, $H$ is connected, and $G/H$ has two irreducible summands. Here, we classify all such spaces and we provide solutions to the so-called Prescribed Ricci…

As is very well-known, linearisation of the instanton equations on a 4-manifold gives rise to an elliptic complex of differential operators, the truncated (twisted) Hodge complex $\Lambda^0(\mathfrak{g}) \to \Lambda^1(\mathfrak{g})\to…

We prove that a compact, intrinsically symmetric submanifold of a Euclidean space is extrinsically symmetric if and only if its maximal tori are Clifford tori in the ambient space. Moreover, we show that this result can be used to give a…

The quaternionic unit ball carries a Riemannian metric built using regular M\"obius transformations: the slice Riemannian metric. We prove that the geometry induced by this metric is strongly related to the group $\mathrm{Sp}(1,1)$. We also…

Logarithmic and $b$-tangent bundles provide a versatile framework for addressing singularities in geometry. Introduced by Deligne and Melrose, these modified bundles resolve singularities by reframing singular vector fields as well-behaved…

In this paper, we demonstrate that on an almost Hermitian manifold $(M^{2n}, J, ds^2)$, a 2-form $\varphi=S^*\Phi$, the pulling back of the K\"ahler form $\Phi$ on the twistor bundle over $M^{2n}$, is non-degenerate if the squared norm…

This paper delves into the concept of ``fat bundles'' within Riemannian submersions. One explores the structural implications of fat Riemannian submersions, particularly focusing on those with non-negative sectional curvature. The main…

We prove the existence of compact surfaces with prescribed constant mean curvature in asymptotically flat and asymptotically hyperbolic manifolds. More precisely, let $(M^3,g)$ be an asymptotically flat manifold with scalar curvature $R\ge…

We consider an invariant gradient flow for the invariant length functional for co-compact curves in inversive geometry, and prove that solutions exist for all time and converge to loxodromic curves, provided the initial curve is admissible…

The fixed points of the generalized Ricci flow are the Bismut Ricci flat metrics, i.e., a generalized metric $(g,H)$ on a manifold $M$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, such that $H$ is $g$-harmonic and…

For a compact K\"ahler manifold, it is well-established that its de Rham cohomology satisfies the Hard Lefschetz condition, which is reflected in the equality between the Betti numbers and the Hodge numbers. A special subclass of symplectic…

Given a compact, connected, and oriented manifold with boundary $M$ and a sequence of smooth Riemannian metrics defined on it, $g_j$, we prove volume preserving intrinsic flat convergence of the sequence to the smooth Riemannian metric…

A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the…