arXiv research

Extends weak continuity of Yang-Mills connections to a broader class.

Classifies compact spaces by shape, finite spaces by weak homotopy.

We provide a direct proof of time-slice weak compactness along the Kähler Ricci flow on Fano manifolds.

This paper is devoted to give a complete unified study of several weak forms of $\ddb-$Lemma on compact complex manifolds.

Study proves existence of weak mean curvature flow with contact angle.

In this note, we prove the existence of weak solutions of the Chern-Ricci flow through blow downs of exceptional curves, as well as backwards smooth convergence away from the exceptional curves on compact complex surfaces. The smoothing property for the Chern-Ricci flow is also obtained on compact Hermitian manifolds o…

Paper proves non-compact inaudibility of symmetry and commutativity.

Study on fourth order Lamm-Riviere system for biharmonic mappings in 4D.

Transcendental holomorphic Morse inequalities aim at characterizing the positivity of transcendental cohomology classes of type $(1,1)$. In this paper, we prove a weak version of Demailly's conjecture on transcendental Morse inequalities on compact Kähler manifolds. And as a consequence, we partially improve a result o…

We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian m…

We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div-curl lemma for tensor fields on Riemannian manifolds is ob…

The paper shows how MMD metrizes weak convergence for certain kernels.

We prove the existence of weak solutions of complex $m-$Hessian equations on compact Hermitian manifolds for the nonnegative right hand side belonging to $L^p, p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has been recently solved by Szekelyhidi and Zhang. We also give a stabilit…

We show a general existence theorem to the complex Monge-Ampère type equation on compact Kähler manifolds.

Study on Kähler manifolds proves weak decompositions and relates harmonic forms.

Sharp regularity for Pfaff system leads to isometric immersions in arbitrary dimensions.

We prove the Focal Index Lemma and the Rauch and Berger comparison theorems on a weak Riemannian Hilbert manifold with a smooth Levi-Civita connection and we apply these results to the free loop spaces of a compact manifold with the L^2 metrics

We discuss pluripotential aspects of the Monge-Ampère equations on compact Hermitian manifolds and prove $L^{\infty}$ estimates for any metric, as well as the existence of weak solutions under an extra assumption.

It is known that the totally umbilical hypersurfaces in the (n+1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S^{n+1}, different from an Euclidean sphere, must have stability index greater than or equa…

In this note we prove that, under a weak condition, small deformations of a compact balanced manifold are also balanced. This condition is satisfied on the twistor space over a compact self-dual four manifold.

In this paper we introduce a general notion of weak extension property for embeddings induced by a group actions. As an example, for the group H(M, m) of measure-preserving homeomorphisms of a noncompact manifold M, we deduce weak type extension theorems, and as an application we exhibit the local contractibility of th…

Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.

We investigate a parabolic-elliptic system which is related to a harmonic map from a compact Riemann surface with a smooth boundary into a Lorentzian manifold with a warped product metric. We prove that there exists a unique global weak solution for this system which is regular except for at most finitely many singular…

New formulations for Ricci flows without smoothness.

Develops weak formulation for spacelike flows in pseudo-Euclidean space.

We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform $L^p$-bounded solution sequence for $p>2$, which implies that the weak limit of the isometric embeddings of the manifold is still an isometric embedding. More generally, we establish a compensated comp…

G2-manifolds with a cohomogeneity-one action of a compact Lie group G are studied. For G simple, all solutions with holonomy G2 and weak holonomy G2 are classified. The holonomy G2 solutions are necessarily Ricci-flat and there is a one-parameter family with SU(3)-symmetry. The weak holonomy G2 solutions are Einstein o…

Weak base-point freeness leads to Kähler-Ricci flow diameter bounds.

The study proves a theorem on Riemannian manifolds for wedge products of weakly convergent differential forms.

Suppose (X,ω) is a compact Kähler manifold. In the present work we propose a simple construction for weak geodesic rays in the space of Kähler metrics that seems to be tied together with properties of the class E(X,ω). As an application of our construction, we prove a characterization of E(X,ω) in terms of envelopes.

Study weak $f$-K-contact manifolds, finding Einstein-type metrics and solitons.

The paper studies special crystallizations of 4-manifolds to minimize certain PL-invariants.

Paper generalizes Andreev's theorem with obtuse angles.

In this paper, we consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra $\mathfrak{g}$, which can be viewed as the extension of Landau-Lifshtiz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the wor…

Polyhedra volume conjecture supports Stoker conjecture weakly.

A new definition of continuous-time equilibrium controls is introduced. As opposed to the standard definition, which involves a derivative-type operation, the new definition parallels how a discrete-time equilibrium is defined, and allows for unambiguous economic interpretation. The terms "strong equilibria" and "weak …

It is well known that in compact local Lipschitz neighborhood retracts in Euclidean space flat convergence for integer rectifiable currents amounts just to weak convergence. In the present paper we extend this result to integral currents in complete metric spaces admitting a local cone type inequality. These include in…

We prove that compact quaternionic-Kähler manifolds of positive scalar curvature admit no almost complex structure, even in the weak sense, except for the complex Grassmannians $Gr_2(C^{n+2})$. We also prove that irreducible inner symmetric spaces $M^{4n}$ of compact type are not weakly complex, except for spheres and …

In this note, we prove that any non-collapsing and compact Gromov-Hausdorff limit of Kahler-Einstein manifolds is either smooth or is orbifold outside a subvariety of complex codimension at least 3.

The main result asserts the existence of continuous solutions of the complex Monge-Ampère equation with the right hand side in $L^p, p>1$, on compact Hermitian manifolds.

Proves higher regularity for anisotropic inverse mean curvature flow.

Study on the geometry of spacelike hypersurfaces in spacetime.

The purpose of this article is to prove existence of mass minimizing integral currents with prescribed possibly non-compact boundary in all dual Banach spaces and furthermore in certain spaces without linear structure, such as injective metric spaces and Hadamard spaces. We furthermore prove a weak$^*$-compactness theo…

Survey on recent developments in isometric immersions using PDE techniques.

Extends mapping results to non-compact Riemannian manifolds with positive reach.

Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampére equations. The purpose of this article, the second of a series on this subject, is to develop a viscosity theory for degenerate complex Mong…

We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular att…

In this paper we generalize the Local Removable Singularity Theorem in [16] for minimal laminations to the case of weak $H$-laminations (with $H\in \mathbb{R}$ constant) in a punctured ball of a Riemannian three-manifold. We also obtain a curvature estimate for any weak CMC foliation (with possibly varying constant mea…