Related papers: A survey on the DDVV conjecture
In this note, we derive an approximation for the mean curvature normal vector on vertices of triangulated surface meshes from the Young-Laplace equation and the force balance principle. We then demonstrate that the approximation expression…
This paper is a sequel to [3]. We formulate a natural algebraic geometry conjecture, give some of its number theoretic and analytical consequences, and show that those can be used to get further advances in wave turbulence theory.
We develop a global theory for complete hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces in…
A conjecture regarding the structure of expander graphs is discussed.
This survey article is about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of…
In this article, we investigate Serrano's conjecture for strictly nef divisors on projective bundles over higher dimensional smooth projective varieties.
On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. We prove that this result may be localized to compact…
We study convex entire graphs evolving with normal velocity equal to a positive power of the mean curvature. Under mild assumptions we prove longtime existence.
We prove the $p$-curvature conjecture for rank two vector bundles with connection on generic curves, by combining deformation techniques for families of varieties and topological arguments.
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…
Following ideas of Gromov we prove scalar and mean curvature comparison results for Riemannian bands with lower scalar curvature bounds in dimension $n\leq7$. The model spaces we use are warped products over scalar-flat manifolds with…
The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme $G$ over a regular local ring $R$ is trivial. The mixed characteristic case of the conjecture is widely open. We consider the…
A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important…
The main purpose of this article is to study higher power mean values of generalized quadratic Gauss sums using estimates for character sums, analytic method and algebraic geometric methods. In this article, we prove two conjectures which…
We prove a formula which compares intersection numbers of conormal varieties of two projective varieties and their dual varieties. When one of them is linear, we can recover the usual Plucker formula for the degree of the dual variety. The…
Comparisons on $L^{n\over 2}$-norms of scalar curvatures between Riemannian metrics and standard metrics are obtained. The metrics are restricted to conformal classes or under certain curvature conditions.
Advection and mean curvature flow is used as a model of bone microarchitecture adaptation. It is an equivalent geometric flow to prescribed mean curvature flow with an additional rate term. In order to validate numerical methods for…
We obtain new average results on the conjectures of Lang-Trotter and Sato-Tate about elliptic curves.
The combination of words ``discrete curvature'' is only an apparent contradiction. In this survey we describe curvature notions associated with polygons, polyhedral surfaces, and with abstract polyhedral manifolds. Several theorems about…
We study the relationship between discrete analogues of Ricci and scalar curvature that are defined for point clouds and graphs. In the discrete setting, Ricci curvature is replaced by Ollivier-Ricci curvature. Scalar curvature can be…