Surface Topology and Geometry

3D dynamic surface tracking is an important research problem and plays a vital role in many computer vision and medical imaging applications. However, it is still challenging to efficiently register surface sequences which has large deformations and strong noise. In this paper, we propose a novel automatic method for non-rigid 3D dynamic surface tracking with surface Ricci flow and Teichmüller map methods. According to quasi-conformal Teichmüller theory, the Techmüller map minimizes the maximal dilation so that our method is able to automatically register surfaces with large deformations. Besides, the adoption of Delaunay triangulation and quadrilateral meshes makes our method applicable to low quality meshes. In our work, the 3D dynamic surfaces are acquired by a high speed 3D scanner. We first identified sparse surface features using machine learning methods in the texture space. Then we assign landmark features with different curvature settings and the Riemannian metric of the surface is computed by the dynamic Ricci flow method, such that all the curvatures are concentrated on the feature points and the surface is flat everywhere else. The registration among frames is computed by the Teichmüller mappings, which aligns the feature points with least angle distortions. We apply our new method to multiple sequences of 3D facial surfaces with large expression deformations and compare them with two other state-of-the-art tracking methods. The effectiveness of our method is demonstrated by the clearly improved accuracy and efficiency.

Feature-aware reconstruction of trimmed splines using Ricci flow with metric optimization

Ruled surfaces, also known as scrolls, are the surfaces built up by moving a straight line along a certain course in the 3D Euclidean space (E³). Many geometers have studied ruled surfaces for centuries to gain rational knowledge about how the surfaces in E³ are generated, and their findings are now essential in all monographs on differential geometry. In the area of optical scanning, scan fields produced by mirror scanning devices of different configurations are surfaces with a light ray as its generating element. This implies, within the accuracy of geometrical optics, that all scan fields are ruled surfaces by nature, and the theory of ruled surfaces in differential geometry provides a land with precious deposits for structural analysis of the scan fields.

String propagation on a curved background defines an embedding problem of surfaces in differential geometry. Using this, we show that in a wide class of backgrounds the classical dynamics of the physical degrees of freedom of the string involves 2-dim sigma-models corresponding to coset conformal field theories.

As a general reference to the topological material covered in this chapter, see Seifert-Threlfall: Lehrbuch der Topologie.

We consider variational inequality problems for set-valued vector fields on general Riemannian manifolds. The existence results of the solution, convexity of the solution set, and the convergence property of the proximal point algorithm for the variational inequality problems for set-valued mappings on Riemannian manifolds are established. Applications to convex optimization problems on Riemannian manifolds are provided.

We study the convergence issue for the gradient algorithm (employing general step sizes) for optimization problems on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity (resp., weak sharp minima), local/global convergence (resp., linear convergence) results are established. As an application, the linear convergence properties of the gradient algorithm employing the constant step sizes and the Armijo step sizes for finding the Riemannian $L^p$ ($p\in[1,+\infty)$) centers of mass are explored, respectively, which in particular extend and/or improve the corresponding results in [B. Afsari, R. Tron, and R. Vidal, SIAM J. Control Optim., 51 (2013), pp. 2230--2260; G. C. Bento et al., J. Optim. Theory Appl., 183 (2019), pp. 977--992].

Gauss as a geometer

We prove the nonlinear local stability of Dirac masses for a kinetic model of alignment of particles on the unit sphere, each point of the unit sphere representing a direction. A population concentrated in a Dirac mass then corresponds to the global alignment of all individuals. The main difficulty of this model is the lack of conserved quantities and the absence of an energy that would decrease for any initial condition. We overcome this difficulty thanks to a functional which is decreasing in time in a neighborhood of any Dirac mass (in the sense of the Wasserstein distance). The results are then extended to the case where the unit sphere is replaced by a general Riemannian manifold.

We study the general solution of the Fokker–Planck equation in d dimensions with arbitrary space and time dependent diffusion matrix and drift terms. We show how to construct the solution for arbitrary initial distributions as an asymptotic expansion for small time. This generalizes the well-known asymptotic expansion of the heat kernel for the Laplace operator on a general Riemannian manifold. We explicitly work out the general solution to leading and next-to-leading order in this small-time expansion, as well as to next-to-next-to-leading order for vanishing drift. We illustrate our results on a several examples.

In this note we introduce and study some new tensors on general Riemannian manifolds which provide a link between the geometry of the underlying manifold and conformally invariant operators (up to order four). We study some of their properties and their relations with well-known geometric objects, such as the scalar curvature, the Q -curvature, the Paneitz operator and the Schouten tensor, and with the elementary conformal tensors \{T^u_{m,\alpha}\} and \{X^u_{m,\mu}\} on Euclidean space introduced in [7] and [6].

In this paper, we study the closed eigenvalue problem of the weighted Paneitz operator and weighted vibration problem for a clamped plateon smooth metric measure spaces, and give the upper bounds for the first $n$ eigenvalues on $n$-dimensional compact submanifolds of a Euclidean space, or a unitsphere, or a projective space, or a general Riemannian manifold. In addition, we give the lower bounds of the first eigenvalue of the weighted vibration problem for a clamped plate on a compact smooth metric measure spacewith the bounded weighted Ricci curvature.

We prove oracle inequalities and upper bounds for kernel density estimators on a very broad class of metric spaces. Precisely we consider the setting of a doubling measure metric space in the presence of a non-negative self-adjoint operator whose heat kernel enjoys Gaussian regularity. Many classical settings like Euclidean spaces, spheres, balls, cubes as well as general Riemannian manifolds, are contained in our framework. Moreover the rate of convergence we achieve is the optimal one in these special cases. Finally we provide the general methodology of constructing the proper kernels when the manifold under study is given and we give precise examples for the case of the sphere.

The Dirichlet-to-Neumann map, viscosity solutions to Eikonal equations, and the self-dual equations of pattern formation

We explain the coarse geometric origin of the fact that certain spectral subspaces of topological insulator Hamiltonians are delocalized, in the sense that they cannot admit an orthonormal basis of localized wavefunctions, with respect to any uniformly discrete set of localization centers. This is a robust result requiring neither spatial homogeneity nor symmetries and applies to Landau levels of disordered quantum Hall systems on general Riemannian manifolds.