Horizontal inverse mean curvature flow in the Heisenberg group

Huisken and Ilmanen in [25] created the theory of weak solutions for inverse mean curvature flows (IMCF) of hypersurfaces on Riemannian manifolds, and proved successfully a Riemannian version of the Penrose inequality. In this paper we investigate and construct the sub-Riemannian version of the theory of weak solutions for inverse mean curvature flows of hypersurfaces in sub-Riemannian Heisenberg groups. We extend the weak solution theory in [25] to the first Heisenberg group and prove the existence, uniqueness and basic geometric properties of horizontal inverse mean curvature flows (HIMCF). By a Heisenberg dilation on HIMCF, we find a horizontal perimeter preserving flow (1.7) in the first Heisenberg group, and prove the existence and uniqueness of weak solutions to (1.7). Using this existential result, the present paper gives a positive answer to an open problem: Heintze-Karcher type inequality in the Heisenberg group. At the same time, this article also proves a Minkowski type formula in the first Heisenberg group

Michael-Simon type inequalities in hyperbolic space

In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space $\mathbb{H}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li, provided that $M$ is $h$-convex and $f$ is a positive smooth function, where $\lambda^{'}(r)=\rm{cosh}$$r$. In particular, when $f$ is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang. Further, we also establish and confirm a new sharp Michael-Simon inequality for the $k$-th mean curvatures in $\mathbb{H}^{n+1}$ by virtue of the Brendle-Guan-Li's flow, provided that $M$ is $h$-convex and $\Omega$ is the domain enclosed by $M$. In particular, when $f$ is of constant and $k$ is odd, (0.2) is exactly the weighted Alexandrov-Fenchel inequalities proven by Hu, Li, and Wei.Comment: 13 page

Equivalence of Strong Brunn-Minkowski Inequalities and CD Conditions in Heisenberg Groups

The present paper investigates the sub-Riemannian version of the equivalence between the curvature-dimension conditions and strong Brunn-Minkowski inequalities in the sub-Riemannian Heisenberg group Hn. We adopt the optimal transport and approximation of Hn developed by Ambrosio and Rigot [1] and combine the celebrated works by M. Magnabosco, L. Portinale and T. Rossi [17] to confirm this.Comment: 23 page

The Minkowski problem in Heisenberg groups

As we all know, the Minkowski type problem is the cornerstone of the Brunn-Minkowski theory in Euclidean space. The Heisenberg group as a sub-Riemannian space is the simplest non-Abelian degenerate Riemannian space that is completely different from a Euclidean space. By analogy with the Minkowski type problem in Euclidean space, the Minkowski type problem in Heisenberg groups is still open. In the present paper, we develop for the first time a sub-Riemannian version of Minkowski type problem in the horizontal distributions of Heisenberg groups, and further give a positive answer to this sub-Riemannian Minkowski type problem via the variational method.Comment: 24 page

An inverse Gauss curvature flow and its application to p-capacitary Orlicz-Minkowski problem

In [Calc. Var., 57:5 (2018)], Hong-Ye-Zhang proposed the $p$-capacitary Orlicz-Minkowski problem and proved the existence of convex solutions to this problem by variational method for $p\in(1,n)$. However, the smoothness and uniqueness of solutions are still open. Notice that the $p$-capacitary Orlicz-Minkowski problem can be converted equivalently to a Monge-Amp\`{e}re type equation in smooth case: \begin{align}\label{0.1} f\phi(h_K)|\nabla\Psi|^p=\tau G \end{align} for $p\in(1,n)$ and some constant $\tau>0$, where $f$ is a positive function defined on the unit sphere $\mathcal{S}^{n-1}$, $\phi$ is a continuous positive function defined in $(0,+\infty)$, and $G$ is the Gauss curvature. In this paper, we confirm the existence of smooth solutions to $p$-capacitary Orlicz-Minkowski problem with $p\in(1,n)$ for the first time by a class of inverse Gauss curvature flows, which converges smoothly to the solution of Equation (\ref{0.1}). Furthermore, we prove the uniqueness result for Equation (\ref{0.1}) in a special case

Quantile hedging for contingent claims in an uncertain financial environment

This paper first studies the quantile hedging problem of contingent claims in an uncertain market model. A special kind of no-arbitrage, that is, the absence of immediate profit, is characterized. Instead of the traditional no-arbitrage targeting the whole market, the absence of immediate profit depends on the confidence level of the portfolio manager for hedging risk. We prove that the condition of absence of immediate profit holds if and only if the initial price of each risky asset lies between the -optimistic value and -pessimistic value of its discounted price at the end of the period. The bounds of the minimal quantile hedging price are derived under the criterion of no-arbitrage in this paper, that is, the absence of immediate profit. Moreover, numerical experiments are implemented to verify that the condition of absence of immediate profit can be a good substitute for the traditional no-arbitrage, since the latter is difficult to achieve. Thus, it may provide a better principle of pricing due to the flexibility from the optional confidence level for the market participants in the increasingly complex financial market

Non-Cash Risk Measure on Nonconvex Sets

Monetary risk measures defined on a convex set are interpreted as the smallest amount of external cash that must be added to a portfolio to make the portfolio being acceptable. In the present paper, the authors introduce a new concept: non-cash risk measure, which does as a nonconvex risk measure work in a nonconvex set. In addition, the authors arrive at a convex extension of the non-cash risk measure, and offer the relationship between the non-cash risk measure and its extension