Related papers: A Pinching constant for harmonic manifolds

We consider compact Riemannian manifolds whose curvature tensor is pointwise negatively pinched, and improve on the corresponding unstable bunching estimate from Hasselblatt's 1994 paper.

We prove partial and full boundary regularity for manifold constrained $p(x)$-harmonic maps.

We study the transversally harmonic maps between foliated Riemannian manifolds. In particular, we prove that under some curvature conditions, any transversally harmonic map is transversally totally geodesic.

In this note we prove that a four-dimensional compact oriented half-confor\-mally flat Riemannian manifold $M^4$ is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2},$ provided that the sectional curvatures all lie in the interval…

We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete…

A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be K\"ahler when the constant is non-zero and must be Chern flat when the constant is zero. The…

Let $M$ be a complete simply connected manifold which is in addition Gromov hyperbolic, coercive and roughly starlike. For a given harmonic function on $M$, a local Fatou Theorem and a pointwise criteria of non-tangential convergence coming…

A classical result of A.D. Alexandrov states that a connected compact smooth $n-$dimensional manifold without boundary, embedded in $\Bbb R^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of…

We investigate the geometry and topology of submanifolds under a sharp pinching condition involving extrinsic invariants like the mean curvature and the length of the second fundamental form. Several homology vanishing results are given.…

We derive monotone properties of positive harmonic functions on three dimensional manifolds with nonnegative scalar curvature, with an asymptotically flat end. Rigidity characterization of spatial Schwarzschild manifolds with two ends is…

Let $M$ be a $C^2$-smooth Riemannian manifold with boundary and $N$ a complete $C^2$-smooth Riemannian manifold. We show that each stationary $p$-harmonic mapping $u\colon M\to N$, whose image lies in a compact subset of $N$, is locally…

We will show that a statistical manifold $(M, g, \nabla)$ has a constant curvature if and only if it is a projectively flat conjugate symmetric manifold, that is, the affine connection $\nabla$ is projectively flat and the curvatures…

We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some…

Brighton [Bri13] proved the Liouville theorem for bounded harmonic functions on weighted manifolds satisfying non-negative curvature dimension condition, i.e. $CD(0,\infty).$ In this paper, we provide a new proof of this result by using the…

In this paper, we prove the Lipschitz regularity of continuous harmonic maps from an finite dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in \cite{lin97}. The proof extends the…

Exploiting the deformation method introduced by Aubin in his seminal work to construct constant negative scalar curvature metrics, we show the existence, on every closed manifold of dimension four, of a metric whose Bach tensor is pinched…

In the biharmonic submanifolds theory there is a generalized Chen's conjecture which states that biharmonic submanifolds in a Riemannian manifold with non-positive sectional curvature must be minimal. This conjecture turned out false by a…

We establish a principle of forced geometric irreducibility on product manifolds. We prove that for any product manifold $M=M_1\times M_2$, a cohomologically calibrated affine connection, $\nabla^{\mathcal{C}}$, is necessarily holonomically…

In the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about constant sectional curvature is replaced by the assumption that the…

Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature, provided M is asymptotically harmonic of constant h > 0.