On Steinerberger Curvature and Graph Distance Matrices

Steinerberger proposed a notion of curvature on graphs (J. Graph Theory, 2023). We show that nonnegative curvature is almost preserved under three graph operations. We characterize the distance matrix and its null space after adding an edge between two graphs. Let $D$ be a graph distance matrix and $\mathbf{1}$ be the all-one vector. We provide a way to construct graphs so that the linear system $Dx = \mathbf{1}$ does not have a solution. Let $\eta$ be the Perron eigenvector of $D.$ We provide a lower bound to $\langle\eta,\mathbf{1}\rangle$ when the graph is a tree.Comment: 3 figure

A New Monotone Quantity in Mean Curvature Flow Implying Sharp Homotopic Criteria

A new monotone quantity in graphical mean curvature flows of higher codimensions is identified in this work. The submanifold deformed by the mean curvature flow is the graph of a map between Riemannian manifolds, and the quantity is monotone increasing under the area-decreasing condition of the map. The flow provides a natural homotopy of the corresponding map and leads to sharp criteria regarding the homotopic class of maps between complex projective spaces, and maps from spheres to complex projective spaces, among others.Comment: 21 page

Entire solutions of two-convex Lagrangian mean curvature flows

Given an entire $C^2$ function $u$ on $\mathbb{R}^n$, we consider the graph of $D u$ as a Lagrangian submanifold of $\mathbb{R}^{2n}$, and deform it by the mean curvature flow in $\mathbb{R}^{2n}$. This leads to the special Lagrangian evolution equation, a fully nonlinear Hessian type PDE. We prove long-time existence and convergence results under a 2-positivity assumption of $(I+(D^2 u)^2)^{-1}D^2 u$. Such results were previously known only under the stronger assumption of positivity of $D^2 u$.Comment: 21 page

A Bernstein type result for special Lagrangian submanifolds

Abstract. Let Σ be a complete minimal Lagrangian submanifold of C n . We identify several regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of Σ lies in one of these regions, then Σ is an affine space

Soliton solutions for the Laplacian coflow of some G2G_2-structures with symmetry

We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt} \psi = - \Delta_d \psi$ where $\psi$ is the dual 4-form of a $G_2$-structure $\phi$ and $\Delta_d$ is the Hodge Laplacian on forms. This flow preserves the condition of the $G_2$-structure being coclosed ($d\psi =0$). We study this flow for two explicit examples of coclosed $G_2$-structures with symmetry. These are given by warped products of an interval or a circle with a compact 6-manifold $N$ which is taken to be either a nearly K\"ahler manifold or a Calabi-Yau manifold. In both cases, we derive the flow equations and also the equations for soliton solutions. In the Calabi-Yau case, we find all the soliton solutions explicitly. In the nearly K\"ahler case, we find several special soliton solutions, and reduce the general problem to a single \emph{third order} highly nonlinear ordinary differential equation.Comment: 18 pages, no figures. Version 2: Minor improvements and addition of references. To appear in Differential Geometry and its Application