Related papers: Rigidity theorem for a static triple with half har…

In this paper, we study scalar curvature rigidity of non-smooth metrics on smooth manifolds with non-positive Yamabe invariant. We prove that if the scalar curvature is not less than the Yamabe invariant in distributional sense, then the…

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is…

In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture.

Extending Aubin's construction of metrics with constant negative scalar curvature, we prove that every $n$-dimensional closed manifold admits a Riemannian metric with constant negative scalar-Weyl curvature, that is $R+t|W|,…

We consider closed biharmonic hypersurfaces in the Euclidean sphere and prove a rigidity result under a suitable condition on the scalar curvature. Moreover, we establish an integral formula involving the position vector for biharmonic…

A quantitative test for the validity of the semi-classical approximation in gravity is given. The criterion proposed is that solutions to the semi-classical Einstein equations should be stable to linearized perturbations, in the sense that…

We prove that an n($\geq$ 4)-dimensional compact Bach-flat manifold with positive constant $\sigma_2$ is an Einstein manifold, provided that its Weyl curvature satisfies a suitable pinching condition.

This paper presents three aspects by which the Weyl geometric generalization of Riemannian geometry, and of Einstein gravity, sheds light on actual questions of physics and its philosophical reflection. After introducing the theory's…

In this paper we prove that the so--called entropy equation, i.e., \[ H\left(x, y, z\right)=H\left(x+y, 0, z\right)+H\left(x, y, 0\right) \] is stable in the sense of Hyers and Ulam on the positive cone of $\mathbb{R}^{3}$, assuming that…

On a compact three-dimensional Riemannian manifold with boundary, we prove the compactness of the full set of conformal metrics with positive constant scalar curvature and constant mean curvature on the boundary. This involves a blow-up…

We establish a homotopy-theoretic description of the homology of stable moduli spaces of $(2n+1)$-dimensional manifold triads $(N, \partial^h N, \partial^v N)$ with fixed $\partial^v N$, whenever $n \geq 3$ and $(N, \partial^h N)$ is…

In this paper, we study semilinear elliptic systems with critical nonlinearity of the form \begin{equation}\label{sys01} \Delta u=Q(x, u, \nabla u), \end{equation} for $u: \mathbb{R}^n\rightarrow \mathbb{R}^K$, $Q$ has quadratic growth in…

We extend to Minkowski spaces the classical result of Barbosa and do Carmo [1] that characterizes the euclidean sphere as the unique compact stable CMC hypersurface of $\mathbb R^n$. More precisely, if $K$ is a smooth convex body in…

We establish integral curvature estimates for complete gradient shrinking Sasaki-Ricci solitons. As an application, we show that any such soliton with harmonic Weyl tensor must be a finite quotient of a sphere. This result can be regarded…

We investigate the interaction between systolic geometry and positive scalar curvature through spinorial methods. Our main theorem establishes an upper bound for the two-dimensional stable systole on certain high-dimensional manifolds with…

Motivated by a question of Tsai-Tsui-Wang, we consider the rigidity of map from manifolds with positive Ricci curvature to manifolds with positive sectional curvature. We show that if the Ricci curvature of the domain dominates that of the…

It is proved, that if a quasi-K\"ahler manifold $M$ of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature $\nu$, then $\nu$, the scalar curvature and the $*$-scalar curvature of $M$ are constants.

We consider surfaces with boundary satisfying a sixth order nonlinear elliptic partial differential equation corresponding to extremising the $L^2$-norm of the gradient of the mean curvature. We show that such surfaces with small $L^2$-norm…

Consider a compact Riemannian manifold M of dimension n whose boundary \partial M is totally geodesic and is isometric to the standard sphere S^{n-1}. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least…

The classical Llarull theorem states that a smooth metric on $n$-sphere cannot have scalar curvature no less than $n(n-1)$ and dominate the standard spherical metric at the same time unless it is the standard spherical metric. In this work,…