arXiv research

Study on 4D solitons with curvature constraints.

Study on manifolds with density using modified Hessians for curvature comparison.

We prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition: their curvature operator can be modified with a 4-form to become positive-semidefinite.

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modifie…

In this paper we define the bi-orthogonal sectional curvature and we present two modified Yamabe invariants for compact 4-dimensional manifolds. In particular we obtained a relationship between one of these invariants and a Hopf conjecture.

In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of Cartan-Hadamard, Synge, and Bonnet-Myers as well as a generalization of the (non-smooth)…

The study proves inequalities and curvature properties for Markov chains.

In this paper, we derive a partial result related to a question of Yau: "Does a simply-connected complete Kähler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?" Main Theorem. Let $M^{2n}$ be a simply-connected complete Kähler manifold M with negative sectional curvature …

Constructs conformal metrics with negative curvature on manifolds with boundary.

In this paper, with the aim of establishing a structure theorem for a compact Kähler manifold $X$ with semi-positive holomorphic sectional curvature, we study a morphism $φ: X \to Y$ to a compact Kähler manifold $Y$ with pseudo-effective canonical bundle. We prove that the morphism $φ$ is always smooth (that is, a subm…

We show, by modifying Borbély's example, that there are $3$-dimen\-sional Cartan-Hadamard manifolds $M$, with sectional curvatures $\le -1$, such that the asymptotic Dirichlet problem for a class of quasilinear elliptic PDEs, including the minimal graph equation, is not solvable.

The paper studies Finsler manifolds with a new curvature concept.

We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an application, we describe an explicit continuous family of ten-dimensional Einst…

In this paper, we derive a sub-gradient estimate for pseudoharmonic maps from noncompact complete Sasakian manifolds which satisfy CR sub-Laplace comparison property, to simply-connected Riemannian manifolds with nonpositive sectional curvature. As its application, we obtain some Liouville theorems for pseudoharmonic m…

New curvature bounds defined for Lorentzian spaces.

In this paper we provide an extension to the Jellett-Minkowski's formula for immersed submanifolds into ambient manifolds which possesses a pole and radial curvatures bounded from above or below by the radial sectional curvatures of a rotationally symmetric model space. Using this Jellett-Minkowski's generalized formul…

Researchers provide counterexamples to Pogorelov and Toponogov's questions about saddle surfaces.

In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from the $L^2$ metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen's determinant line bundle…

Study a specific line arrangement and compute its fundamental group via braid monodromy.

The Chern sectional curvature of a Hermitian manifold is derived and related to Kähler metrics.

New proof shows holomorphic sectional curvature fully determines curvature tensor.

A semi-Riemannian manifold is said to satisfy $R\ge K$ (or $R\le K$) if spacelike sectional curvatures are $\ge K$ and timelike ones are $\le K$ (or the reverse). Such spaces are abundant, as warped product constructions show; they include, in particular, big bang Robertson-Walker spaces. By stability, there are many n…

Smooth solutions found for modified mean curvature flow in Riemannian manifolds.

The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.

We prove that a product complex manifold cannot admit a complete Kähler metric with sectional curvature $K<c<0$ and Ricci curvature $Ric > d$, where $c$ and $d$ are constants. In particular, a product domain in $\C$ cannot cover a compact Kähler manifold with negative sectional curvature. On the other hand, we observe …

This paper is about geometric quantization of the Hitchin system. We quantize a Kahler form on the Hitchin moduli space (which is half the first Kahler form defined by Hitchin) by considering the Quillen bundle as the prequantum line bundle and modifying the Quillen metric using the Higgs field so that the curvature is…

The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.

Smooth convergence to equidistant surfaces in Fuchsian manifolds.

We prove the following results: An almost Hermitian manifold of indefinite metric is of pointwise constant holomorphic sectional curvature if the holomorphic sectional curvature is bounded from above and from below. If the antiholomorphic sectional curvature is bounded either from above or from below, then the manifold…

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative c…

It is proved that if an AK2-manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then it is a 6-dimensional manifold of constant negative sectional curvature or a Kähler manifold of constant holomorphic sectional curvature.

Formula for sectional curvatures on matrix groups.

Formula for sectional curvature on 2D Lorentzian manifolds derived.

The φ-sectional curvature of statistical structures on almost contact metric manifolds is always non-positive.

4-manifolds with nonnegative sectional curvature are area-extremal.

The paper extends Gray's result to quaternion-Kähler manifolds.

In this paper, we obtain classification of four-dimensional Einstein manifolds with positive Ricci curvature and pinched sectional curvature. In particular, the first result concerns with an upper bound of sectional curvature, improving a theorem of E. Costa. The second is a generalization of D. Yang's result assuming …

Curvature flows in hyperbolic space preserve positive sectional curvature and contract to a point.

The paper proves properties of complex manifolds with nonnegative holomorphic sectional curvature.

New theorem links quaternionic-Kähler manifolds to symmetric spaces.

Let M be an almost Hermitian manifold of dimension greater or equal to 6. The following theorems are proved: Theorem 1. If M is of pointwise constant θ-holomorphic sectional curvature for a number θ in (0,π/2) then M is of constant sectional curvature or a Kähler manifold of constant holomorphic sectional curvature. Th…

In dimension greater than four, we prove that if a Hermitian non-Kaehler manifold is of pointwise constant antiholomorphic sectional curvatures, then it is of constant sectional curvatures.

Global and local estimates for a curvature equation on manifolds with boundary.

Directly proves Wu's theorem on negative curvature metrics.

Ricci flow preserves positive sectional curvature on homogeneous spheres

Study non-negative curvature Markov chains, proving entropy contraction.

No conformal product structures on compact manifolds with constant curvature.

In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field…