相关论文: PL Morse theory in low dimensions
Branch points of a real 2-surface S in a 4-manifold M generalize the branch points of complex curves in complex surfaces: for example, they can occur as singularities of minimal surfaces. We investigate such a branch point p when S is…
In this paper, we will study the partial regularity theorem for stationary harmonic maps from a Riemannian manifold into a Lorentzian manifold. For a weakly stationary harmonic map $(u,v)$ from a smooth bounded open domain…
This paper, a culmination of the authors' theory of the RT-equations, accomplishes the following: (i) We discover there is a true (geometric) regularity associated with every affine connection, its ``essential regularity'', the highest…
Building upon dimension reduction techniques in the study of positive scalar curvature (PSC) geometry, we prove an effective version of the positive mass theorem (PMT) for asymptotically flat (AF) manifolds of dimension $n\leq 8$ with…
It is conjectured since long that for any convex body $P\subset \mathbb{R}^n$ there exists a point in its interior which belongs to at least $2n$ normals from different points on the boundary of $P$. The conjecture is known to be true for…
We carry out a detailed quantitative analysis on the geometry of invariant manifolds for smooth dissipative systems in dimension two. We begin by quantifying the regularity of any orbit (finite or infinite) in the phase space with a set of…
On a smooth manifold, we associate to any closed differential form a mapping cone complex. The cohomology of this mapping cone complex can vary with the de Rham cohomology class of the closed form. We present a novel Morse theoretical…
We draw elliptic regularity results for 4-manifolds with an elliptic system, without Sobolev constant control. Direct use of analysis is circumvented; the results come mainly through geometric and topological arguments. In contrast to our…
Persistent Homology (PH) allows tracking homology features like loops, holes and their higher-dimensional analogs, along with a single-parameter family of nested spaces. Currently, computing descriptors for complex data characterized by…
One of the basic aims of this paper is to study the relationship between the geometry of ``hypersurface like'' subsets of Euclidean space and the properties of the measures they support. In this context we show that certain doubling…
We give a necessary and sufficient geometric structural condition for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable…
We examine domain-valued maxitive measures defined on the Borel subsets of a topological space. Several characterizations of regularity of maxitive measures are proved, depending on the structure of the topological space. Since every…
We prove a result analogous to Reeb's theorem in the context of Morse-Bott functions: if a closed, smooth manifold $M$ admits a Morse-Bott function having two critical submanifolds $S^k$ and $S^l$ ($k \neq l$), then $M$ has dimension…
Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f: O -> P between smooth orbifolds O and P. We show that Sard's theorem holds and that the inverse image of a regular value is a…
Numerous structural findings of homology manifolds have been derived in various ways in relation to $g_2$-values. The homology $4$-manifolds with $g_2\leq 5$ are characterized combinatorially in this article. It is well-known that all…
We study the second Huber theorem in dimensions 2 and 4. In dimension 2, we prove a new version assuming that the Gauss curvature lies in a negative Sobolev space using Coulomb frames. In dimension $4$, given a metric having a pointwise…
We ask whether a stationary lattice in dimension $d$ whose points are shifted by identically distributed but possibly dependent perturbations remains hyperuniform. When $d = 1$ or $2$, we show that it is the case when the perturbations have…
We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of $\mathbb{R}^n$, under compactly supported variations. The critical point solves a fourth order…
Special generic maps are smooth maps at each singular point of which we can represent as $(x_1, \cdots, x_m) \mapsto (x_1,\cdots,x_{n-1},\sum_{k=n}^{m}{x_k}^2)$ for suitable coordinates. Morse functions with exactly two singular points on…
A manifestly supersymmetric nonperturbative matrix regularization for a twisted version of N=(8,8) theory on a curved background (a two-sphere) is constructed. Both continuum and the matrix regularization respect four exact scalar…