We describe a new class of holonomy groups on pseudo-Riemannian manifolds. Namely, let $g$ be a nondegenerate bilinear form on a vector space $V$, and $L : V \to V$ a $g$-symmetric operator. Then the identity component of the centraliser of $L$ in $\mathrm{SO}(g)$ is a holonomy group for a suitable Levi-Civita connection.

In this paper, we address the Calabi-Lee conjecture for pseudo-Einstein contact structure via the CR Poincaré-Lelong equation. Then we confirm the Calabi-Yau Theorem via Hodge-Laplacian heat flow in a closed strictly pseudoconvex CR $(2n + 1)$-manifold $(M , \theta)$ for $n \geq 2$. With its applications, we affirm a partial answer of the CR Frankel conjecture in a closed spherical strictly pseudoconvex CR $(2n + 1)$-manifold.

The Orlicz-Brunn-Minkowski theory, introduced by Lutwak, Yang, and Zhang, is a new extension of the classical Brunn-Minkowski theory. It represents a generalization of the $L_p$-Brunn-Minkowski theory, analogous to the way that Orlicz spaces generalize $L_p$ spaces. For appropriate convex functions $\varphi : [0,\infty)^m \to [0,\infty)$, a new way of combining arbitrary sets in $\mathbb{R}^n$ is introduced. This operation, called Orlicz addition and denoted by ${+}_{\varphi}$, has several desirable properties, but is not associative unless it reduces to $L_p$ addition. A general framework is introduced for the Orlicz-Brunn-Minkowski theory that includes both the new addition and previously introduced concepts, and makes clear for the first time the relation to Orlicz spaces and norms. It is also shown that Orlicz addition is intimately related to a natural and fundamental generalization of Minkowski addition called $M$-addition. The results obtained show, roughly speaking, that the Orlicz-Brunn-Minkowski theory is the most general possible based on an addition that retains all the basic geometrical properties enjoyed by the $L_p$-Brunn-Minkowski theory.

Inequalities of the Brunn-Minkowski type are obtained, both for M-addition and Orlicz addition. The new Orlicz-Brunn-Minkowski inequality implies the $L_p$-Brunn-Minkowski inequality. New Orlicz-Minkowski inequalities are obtained that generalize the $L_p$-Minkowski inequality. One of these has connections with the conjectured log-Brunn-Minkowski inequality of Lutwak, Yang, and Zhang, and in fact these two inequalities together are shown to split the classical Brunn-Minkowski inequality.

For $2 \leq k \leq d - 1$, the $k$-th mean section body, $M_k(K)$, of a convex body $K$ in $\mathbb{R}^d$, is the Minkowski sum of all its sections by $k$-dimensional flats. We will show that the characterization of these mean section bodies is equivalent to the solution of the general Minkowski problem, namely that of giving the characteristic properties of those measures on the unit sphere which arise as surface area measures (of arbitrary degree) of convex bodies. This equivalence arises from an analysis of Berg's solution of the Christoffel problem. We will see how the functions introduced by Berg yield an integral representation of the support function of $M_k(K)$ in terms of the $(d + 1 - k)$-th surface area measure of $K$. Our results will be obtained using Fourier transform techniques which also yield a stability version of the fact that $M_k(K)$ determines $K$ uniquely.

We prove a bridge principle at infinity for area-minimizing surfaces in the hyperbolic $\mathbb{H}^3$, and we use it to prove that any open, connected, orientable surface can be properly embedded in $\mathbb{H}^3$as an area-minimizing surface. Moreover, the embedding can be constructed in such a way that the limit sets of different ends are disjoint.

We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space.We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows us to smoothly flow through singularities by studying graphical mean curvature flow in one additional dimension.