arXiv research

New proof of Kobayashi's properness criterion using metric geometry.

Criterion for Lie algebroid connections on compact Riemann surfaces.

Criterion for solvability of complex 2-Hessian equation on compact Kähler manifolds.

The paper studies the modified J-equation on Kähler manifolds.

In this paper, we first get a criterion formula for whether a differential form is holomorphic with respect to the generalized complex structure induced by $ε$. Next, we get the local extensions of $\overline\partial$-closed forms on a smooth family of compact generalized Hermitian manifolds by using this criterion. Fi…

Study minimizers in large volume isoperimetric problems with a new flatness criterion.

Improved proofs for topological and smooth pseudo-isotopies of simply connected 4-manifolds.

A new construction of a universal connection was given in \cite{BHS}. The main aim here is to explain this construction. A theorem of Atiyah and Weil says that a holomorphic vector bundle $E$ over a compact Riemann surface admits a holomorphic connection if and only if the degree of every direct summand of $E$ is degre…

New invariant for hyperbolic surfaces, geometric criterion for domains.

Characterizes pseudo-Anosov orbit spaces via bifoliated planes

New criterion found for Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.

The paper studies Randers and $(α,β)$ equigeodesics on compact homogeneous manifolds.

Paper bounds surface diameter and solves Plateau-Douglas problem.

We show that any simply connected topological closed $4$-manifold punctured along any compact, totally disconnected tame subset $Λ$ admits a continuum of smoothings which are not diffeomorphic to any leaf of a $C^{1,0}$ codimension one foliation on a compact manifold. This includes the remarkable case of $S^4$ puncture…

We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits $G/K = \varinjlim G_n/K_n$ of compact riemannian symmetric spaces, …

We study quaternionic Bott-Chern cohomology on compact hypercomplex manifolds and adapt some results from complex geometry to the quaternionic setting. For instance, we prove a criterion for the existence of HKT metrics on compact hypercomplex manifolds of real dimension 8 analogous to the one given by Teleman [35] and…

Let $K\backslash G$ be an irreducible Hermitian symmetric space of noncompact type and $Γ\,\subset\, G$ a closed torsionfree discrete subgroup. Let $X$ be a compact Kähler manifold and $ρ\, :\, π_1(X, x_0)\,\longrightarrow\, Γ$ a homomorphism such that the adjoint action of $ρ(π_1(X, x_0))$ on $\text{Lie}(G)$ is comple…

Uniformizes compact Sasakian manifolds into circle bundles.

In the first part of this paper, we consider smooth maps from a compact orientable 3-manifold without boundary to the 2-sphere. We give a geometric criterion to decide whether two given maps are homotopic, based on the sets of points where the maps are equal or antipodal. We extend this criterion to non-singular vector…

Criterion found for Lie algebroid connections on parabolic bundles.

We introduce a new cohomology-theoretic method for classifying generic immersed curves in closed compact surfaces by using Gauss codes. This subsumes a result of J.S. Carter on classifying immersed curves in oriented compact surfaces, and provides a criterion for when an immersion is two-colorable. We note an applicati…

We examine volume pinching problems of CAT(1) spaces. We characterize a class of compact geodesically complete CAT(1) spaces of small specific volume. We prove a sphere theorem for compact CAT(1) homology manifolds of small volume. We also formulate a criterion of manifold recognition for homology manifolds on volume g…

New criteria for non-isometric group actions in metric spaces.

We construct solutions to the constraint equations in general relativity using the limit equation criterion introduced by Dahl, Humbert and the first author. We focus on solutions over compact 3-manifolds admitting a $\bS^1$-symmetry group. When the quotient manifold has genus greater than 2, we obtain strong far from …

We give a necessary and sufficient condition for orbits of commutative Hermann actions and actions of the direct product of two symmetric subgroups on compact Lie groups to be biharmonic in terms of symmetric triad with multiplicities. By this criterion, we determine all the proper biharmonic submanifolds in irreducibl…

New criterion for generating free groups in CAT(0) spaces.

Proves regularity of extremal function on compact Kähler manifolds.

GCNNs on homogeneous spaces use vector bundles and Hilbert spaces.

Study finds criteria for surfaces with specific curvature properties.

A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups which enables one to decide whether there is a lattice or not. Moreover, it is eas…

We present a new criterion for the complex hyperbolicity of a non-compact quotient X of a bounded symmetric domain. For each p $\ge$ 1, this criterion gives a precise condition under which the subvarieties V $\subset$ X with dim V $\ge$ p are of general type, and X is p-measure hyperbolic. Then, we give several applica…

In this paper, we give a criterion for the properness of the K-energy in a general Kahler class of a compact Kahler manifold by using Song-Weinkove's result. As applications, we give some Kahler classes on $\mathbb{C}\mathbb{P}^2\#3\overline {\mathbb{C}\mathbb{P}^2}$ and $\mathbb{C}\mathbb{P}^2\#8\overline {\mathbb{C}\…

We present some new Stokes' type theorems on complete non-compact manifolds that extend, in different directions, previous work by Gaffney and Karp and also the so called Kelvin-Nevanlinna-Royden criterion for (p-)parabolicity. Applications to comparison and uniqueness results involving the p-Laplacian are deduced.

Study on stable minimal hypersurfaces under Ricci curvature constraints.

Criterion for polystability in Lie group actions on manifolds.

We consider the Yamabe invariant of a compact orbifold with finitely many singular points. We prove a fundamental inequality for the estimate of the invariant from above, which also includes a criterion for the non-positivity of it. Moreover, we give a sufficient condition for the equality in the inequality. In order t…

We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve h…

An old theorem of Weil and Kodaira says that for a compact Kähler manifold $X$ there is a closed logarithmic $1$-form with residue divisor $D$ if and only if $D$ is homologous to zero in $H_{2n-2}(X,\mathbb C)$. In the first part of this paper, we generalize the above theorem to general compact complex manifolds by sho…

Joyce's criterion for sLag smoothings extended to non-compact, non-transverse intersections.

We construct Weil-Petersson (WP) geodesic rays with minimal filling non-uniquely ergodic ending lamination which are recurrent to a compact subset of the moduli space of Riemann surfaces. This construction shows that an analogue of the Masur's criterion for Teichmüller geodesics does not hold for WP geodesics.

Compactness theorem for quasiregular curves proves normality and resolves nodal points.

Let $X$ be a compact toric extremal Kähler manifold. Using the work of Székelyhidi, we provide a combinatorial criterion on the fan describing $X$ to ensure the existence of complex deformations of $X$ that carry extremal metrics. As an example, we find new CSC metrics on 4-points blow-ups of $\C¶^1\times\C¶^1$.

The paper shows how compact Kähler manifolds with a special bundle can be broken down into simpler parts.

New Fredholm criteria for pseudodifferential operators on manifolds.

The paper connects curvature positivity to rational connectedness in complex geometry.

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We fin…

Kodaira embedding theorem provides an effective characterization of projectivity of a Kähler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms…

We prove that Demailly's holomorphic Morse inequalities hold true for complex orbifolds by using a heat kernel method. Then we introduce the class of Moishezon orbifolds and as an application of our inequalties, we give a geometric criterion for a compact connected orbifold to be a Moishezon orbifolds, thus generalizin…