arXiv research

The paper studies equations on almost Hermitian manifolds with estimates and existence results.

Over a compact Kähler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a Kähler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature Kähler metrics with conic singularities: existence result under small deformatio…

New methods prove existence of rotating shapes moving in space.

In this work, we obtain a existence criteria for the longtime Kähler Ricci flow solution. Using the existence result, we generalize a result by Wu-Yau on the existence of Kähler Einstein metric to the case with possibly unbounded curvature. Moreover, the Kähler Einstein metric with negative scalar curvture must be uniq…

Study finds existence of $Q$-curvature metrics on even-dimensional manifolds with conical singularities.

Authors review CMC spacelike hypersurfaces and new existence results.

Study existence and uniqueness of solutions for Yamabe problem on non-compact manifolds with negative curvature.

New results on geodesic flows using curve shortening flow.

In this paper we compute the Futaki invariant of adiabatic Kaehler classes on resolutions of Kaehler orbifolds with isolated singularities. Combined with previous existence results of extremal metrics by Arezzo-Lena-Mazzieri, this gives a number of new existence and non-existence results for cscK metrics.

Extends ASD connection existence to 4-manifolds with cylindrical ends.

The paper proves short-time existence and uniqueness of Ricci flow on Finsler manifolds.

Paper explores existence and nonexistence of submanifolds in contact geometry.

We prove an existence result for exact lagrangian cobordisms between closed legendrians.

This paper concerns a fully nonlinear version of the Yamabe problem on manifolds with boundary. We establish some existence results and estimates of solutions.

Study proves short-term existence for harmonic maps under evolving metrics.

Many results in mathematical relativity, including results for both the initial data problem and for the evolution problem, rely on the existence of a constant mean curvature (CMC) Cauchy surface in the underlying spacetime. However, it is known that some spacetimes have no CMC Cauchy surfaces (slices). This is an obst…

Extends rigidity and existence results for discrete conformal structures on surfaces with boundary.

The paper investigates polyharmonic helices in 3D solvable Lie group Sol_3 and Euclidean spheres.

In this paper the fractional Q-curvature problem on three dimensional CR sphere is considered. By using the critical points theory at infinity, an existence result is obtained.

Study shows strict inequality for Dirac-type equation on spin manifolds, leading to existence results.

We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.

Study on surfaces with conical singularities and geodesic boundaries, deriving existence results.

Study proves curvature flow existence on CR manifolds.

Proves existence of certain subgroups in hyperbolic groups.

Paper investigates prescribing Chern scalar curvatures on specific manifolds.

Study on trapped submanifolds in spacetimes, proving rigidity and non-existence results.

Extends existence results for scalar curvature on conical manifolds.

Study anisotropic flows solving Orlicz-Minkowski problems, proving existence and new results.

We prove the existence of minimal hypersurfaces for the Dirichlet that extends a similar result of Jenkins and Serrin in Euclidean Space to Riemannian ambient manifolds

Study of harmonic maps on 2D simplicial complexes, proving existence and regularity.

Study proves existence of closed geodesics on spheres and projective spaces.

We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases.

New CMC existence result for expanding cosmological spacetimes.

Study on prescribing curvature on specific manifolds with negative Gauduchon degree.

Existence and uniqueness of gravitating vortices on Riemann surfaces with specific properties.

Study proves existence of robust classifiers in multiclass adversarial training.

New existence results for curvature problem on balls with specific conditions.

In this paper we prescribe a fourth order conformal invariant on the standard $n-$sphere, with $n\geq5$, and study the related fourth order elliptic equation. We first find some existence results in the perturbative case. After some blow up analysis we build a homotopy to pass from the perturbative case to the non-pert…

We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore theorem that the Finsler geodesic flow can be interpreted as a Reeb flow. As a mil…

In this note, we prove an existence result on exhaustion functions adapting the method by L.-F. Tam. Then we apply it to prove short-time existence of Ricci flow and study Yau's uniformization conjecture using similar method as Fei He and Lee-Tam.

In a recent paper, the authors proved that no spin foliation on a compact enlargeable manifold with Hausdorff homotopy graph admits a metric of positive scalar curvature on its leaves. This result extends groundbreaking results of Lichnerowicz, Gromov and Lawson, and Connes on the non-existence of metrics of positive s…

Prove long-time existence of pluriclosed flow on certain fibrations

In this work, we use the global analysis and degree-theoretic methods introduced by Smale to study the existence and multiplicity of solutions of the vacuum Einstein constraint equations given by the conformal method of Lichnerowicz-Choquet-Bruhat-York. In particular this approach gives a new proof of the existence res…

We prove the global existence of Dirac-wave maps with curvature term with small initial data on globally hyperbolic manifolds of arbitrary dimension which satisfy a suitable growth condition. In addition, we also prove a global existence result for wave maps under similar assumptions.

We consider the fractional Nirenberg problem on the standard sphere $\mathbb{S}^n$ with $n\geq 4$. Using the theory of critical points at infinity, we establish an Euler-Hopf type formula and obtain some existence results for curvature satisfying assumptions of Bahri-Coron type.

We study the steady state solutions of a generalized logistic type equation on a complete Riemannian manifold. We provide sufficient conditions for existence, respectively non-existence of positive solutions, which depend on the relative size of the coefficients and their mutual interaction with the geometry of the man…

We prove some existence and non-existence results for complete area minimizing surfaces in the homogeneous space $\mathbb{E}(-1,τ)$. As one of our main results, we present sufficient conditions for a curve $Γ$ in $\partial_{\infty} \mathbb{E}(-1,τ)$ to admit a solution to the asymptotic Plateau problem, in the sense th…

In this paper, we will study the existence problem of minmax minimal torus. We use classical conformal invariant geometric variational methods. We prove a theorem about the existence of minmax minimal torus in Theorem 5.1. Firstly we prove a strong uniformization result(Proposition 3.1) using method of [1]. Then we use…