Related papers: Positive curvature operator, projective manifold a…

In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min-max theory for the area functional to prove this conjecture in the positive Ricci…

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…

We prove two results on geometric consequences of the representation of restricted holonomy group of a Hermitian connection. The first result concerns when such a Hermitian manifold is K\"ahler in terms of the torsion and the irreducibility…

The goal of this article is to study the pinching problem proposed by S.-T. Yau in 1990 replacing sectional curvature by one weaker condition on biorthogonal curvature. Moreover, we classify 4-dimensional compact oriented Riemannian…

For a smooth compact Riemannian manifold with positive Yamabe invariant, positive Q curvature and dimension at least 5, we prove the existence of a conformal metric with constant Q curvature. Our approach is based on the study of extremal…

Given a (reduced) locally compact quantum group $A$, we can consider the convolution algebra $L^1(A)$ (which can be identified as the predual of the von Neumann algebra form of $A$). It is conjectured that $L^1(A)$ is operator biprojective…

Let X be a complex projective variety of dimension n with only isolated normal singularities. In this paper we prove, using mixed Hodge theory, that if the link of each singular point of X is (n-2)-connected, then X is a formal topological…

This is a slightly altered version of the authors thesis from 2014. In the first main part we show that the quotient space of a compact, simply connected and nonnegatively curved Riemannian 4-manifold by an effective, isometric…

In this paper, we extend a result of Gao Chen regarding the solvability of the twisted deformed Hermitian Yang-Mills equations on compact K\"ahler manifolds to allow for the twisting function to be non-constant and slightly negative in all…

Let $ X $ be an oriented, closed manifold with $ \dim X \geqslant 2 $. Let $ (Z, \partial Z) $ be an oriented, compact manifold with (possibly empty) smooth boundary and $ \dim Z \geqslant 2 $. In this article, we show that if the…

In this article, we classify (non-compact) $3$-manifolds with uniformly positive scalar curvature. Precisely, we show that an oriented $3$-manifold has a complete metric with uniformly positive scalar curvature if and only if it is…

In this note, we introduce a new curvature condition called the $2-$positive bisectional curvature on compact K\"{a}hler manifolds. We then deduce a characterization theorem for manifolds with $2-$positive bisectional curvature, which can…

In this paper we study the relationship between three compactifications of the moduli space of Hermitian-Yang-Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the…

We establish a link between rational homotopy theory and the problem which vector bundles admit complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively…

We investigate a new property for compact Kahler manifolds. Let X be a Kahler manifold of dimension n and let H^{1,1} denote the (1,1) part of its real second cohomology. On this space, we have an degree n form given by cup product. Let K…

We prove the Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds, that is, for projective manifolds equipped with a holomorphic action of a compact Lie group with at least one real hypersurface orbit. Contrary to what seems to be…

In this paper, we study the topological rigidity and its relationship with the positivity of scalar curvature. Precisely, we show that any complete contractible $3$-manifold with non-negative scalar curvature is homeomorphic to…

We give some rigidity theorems for an n$(\geq4)$-dimensional compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $\sigma_2$. Moreover, when $n=4,$ we prove that a 4-dimensional compact…

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

Let $M$ be a complete Ricci-flat Kahler manifold with one end and assume that this end converges at an exponential rate to $[0,\infty) \times X$ for some compact connected Ricci-flat manifold $X$. We begin by proving general structure…