This is an essay on potential theory for geometric plurisubharmonic functions. It begins with a given closed subset G of the Grassmann bundle $G(p,TX)$ of tangent $p$-planes to a riemannian manifold $X$. This determines a nonlinear partial differential equation which is convex but never uniformly elliptic (p < dim X). A surprising number of results in complex analysis carry over to this more general setting. The notions of: a G-submanifold, an upper semi-continuous G-plurisubharmonic function, a G-convex domain, a G-harmonic function, and a G-free submanifold, are defined. Results include a restriction theorem as well as the existence and uniqueness of solutions to the Dirichlet Problem for G-harmonic functions on G-convex domains.
Three results in p-convex geometry are established. First is the analogue of the Levi problem in several complex variables, namely: local p-convexity implies global p-convexity. The second asserts that the support of a minimal p-dimensional current is contained in the p-hull of the boundary union with the "core" of the space. Lastly, the exteme rays in the convex cone of p-positive matrices are characterized. This is a basic result with many applications.
Let $π:\mc{X}\to \mc{T}$ be Teichmüller curve over Teichmüller space $\mc{T}$, such that the fiber $\mc{X}_z=π^{-1}(z)$ is exactly the Riemann surface given by the complex structure $z\in \mc{T}$. For a fixed Riemannian manifold $M$ and a continuous map $u_0: M\to \mc{X}_{z_0}$, let $E(z)$ denote the energy function of the harmonic map $u(z):M\to \mc{X}_z$ homotopic to $u_0$, $z\in \mathcal T$. We obtain the first and the second variations of the energy function $E(z)$, and show that $\log E(z)$ is strictly plurisubharmonic on Teichmüller space, from which we give a new proof on the Steinness …
Recently the authors have explored new concepts of plurisubharmonicity and pseudoconvexity, with much of the attendant analysis, in the context of calibrated manifolds. Here a much broader extension is made. This development covers a wide variety of geometric situations, including, for example, Lagrangian plurisubhamonicity and convexity. It also applies in a number of non-geometric situations. Results include: fundamental properties of $P^+$-plurisubharmonic functions, plurisubharmonic distributions and regularity, $P^+$-convex domains and $P^+$-convex boundaries, topological restrictions on …
In view of A. Andreotti and H. Grauert's vanishing theorem for q-complete domains in C^n, (Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193--259,) we re-prove a vanishing result by J.-P. Sha, (p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437--447,) and H. Wu, (Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525--548,) for the de Rham cohomology of strictly p-convex domains in R^n in the sense of F. R. Harvey and H. B. Lawson, (The foundations of p-convexity and p-plurisubharmonicity in r…
We prove that the static convexity is preserved along two kinds of locally constrained curvature flows in hyperbolic space. Using the static convexity of the flow hypersurfaces, we prove new family of geometric inequalities for such hypersurfaces in hyperbolic space.
In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.
The Hessian of the renormalized volume of geometrically finite hyperbolic $3$-manifolds without rank-$1$ cusps, computed at the hyperbolic metric $g$ with totally geodesic boundary of the convex core, is shown to be a strictly positive bilinear form on the tangent space to Teichmüller space. The metric $g$ is known from results of Bonahon and Storm to be an absolute minimum for the volume of the convex core. We deduce the strict convexity of the functional volume of the convex core at its minimum point.
We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the $k$-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve $k$-convexity in a Riemannia…
In this short paper we introduce a variant of the approach to inverting the X-ray transform that originated in the author's work with Uhlmann. The new method is based on semiclassical analysis and eliminates the need for using sufficiently small domains and layer stripping for obtaining the injectivity and stability results, assuming natural geometric conditions are satisfied.
The main result of this paper is a convexity estimate for translating solitons of extrinsic geometric flows which evolve under a $1$-homogeneous concave function in the principal curvatures. In addition, we show examples of these hypersurfaces in $\mathbb{R}^{n+1}$ for particular functions.
The main goal of this paper is to present results of existence and non-existence of convex functions on Riemannian manifolds and, in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove that the conservativity of the geodesic flow on a Rieman- nain manifold with infinite volume is an obstruction to the existence of convex functions. Next, we present a geometric condition that ensures the existence of (strictly) convex functions on a particular class of complete non-compact man- ifolds, and, we use this fact to construct a manifold whose se…
We establish a geometric lower bound for the principal curvature of the level surfaces of solutions to $F(D^2u, Du, u, x)=0$ in convex ring domains, under a refined structural condition introduced by Bianchini-Longinetti-Salani in \cite{BLS}. We also prove a constant rank theorem for the second fundamental form of the convex level surfaces of these solutions.
In this work, we propose a new evolving geometric flow (called translating mean curvature flow) for the translating solitons of hypersurfaces in $R^{n+1}$. We study the basic properties, such as positivity preserving property, of the translating mean curvature flow. The Dirichlet problem for the graphical translating mean curvature flow is studied and the global existence of the flow and the convergence property are also considered.
In this paper we prove some geometric inequalities for closed surfaces in Euclidean three-space. Motivated by Gage's inequality for convex curves, we first verify that for convex surfaces the Willmore energy is bounded below by some scale-invariant quantities. In particular, we obtain an optimal scaling law between the Willmore energy and the isoperimetric ratio under convexity. In addition, we also address Topping's conjecture relating diameter and mean curvature for connected closed surfaces. We prove this conjecture in the class of simply-connected axisymmetric surfaces, and moreover obtain…
We give a proof of the Donnelly-Fefferman growth bound of Laplace-Beltrami eigenfunctions which is probably the easiest and the most elementary one. Our proof also gives new quantitative geometric estimates in terms of curvature bounds which improve and simplify previous work by Garofalo and Lin. The proof is based on a convexity property of harmonic functions on curved manifolds, generalizing Agmon's Theorem on a convexity property of harmonic functions in R^n.
Every harmonic map is an intrinsic bi-harmonic map as an absolute minimizer of the intrinsic bi-energy functional, therefore intrinsic bi-harmonic map and its heat flow are more geometrically natural to study, but they are also considerably more difficult analytically than the extrinsic counterparts due to the lack of coercivity for the intrinsic bi-energy. In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit $4$-ball $B_1 \subset \mathbf{R}^4$ into the sphere $\mathbf{S}^n$. This is a higher-order analogue of the energy convexity a…
Geometric uncertainty principles for Schrödinger evolutions on negatively curved manifolds
Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；National Key Laboratory of Computational Physics（计算物理国家重点实验室）；The Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）
In this paper we study domains in flag manifolds which are bounded in an affine chart and whose projective automorphism group acts co-compactly. In contrast to the many examples in real projective space, we will show that no examples exist in many flag manifolds. Moreover, in the cases where such domains can exist, we show that they satisfy a natural convexity condition and have an invariant metric which generalizes the Hilbert metric. As an application we give some restrictions on the developing map for certain $(G,X)$-structures.
The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard [Ann. Statist. 47 177-204 (2019)], an estimator for the reach is given. A uniform expected loss bound over a C^k model is found. Lower bounds for the minimax rate for estimating the reach over these models are also provided. The estimator almost …
