arXiv research

Study circle actions on unitary manifolds with discrete fixed points.

The paper classifies circle actions on 6D manifolds with isolated fixed points.

New method upsamples sparse, non-uniform point clouds more accurately.

Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$. Fixed-point homogeneous manifolds with positive sectional curvature have been c…

A Riemannian manifold is said to be uniformly secure if there is a finite number $s$ such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by $s$ point obstacles. We prove that the number of geodesics with length $\leq T$ between every pair of points in a uniformly secure manifol…

Study circle actions on manifolds with 3 fixed points, finding dimension constraints and unique structures.

We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points.

Proof shows volume equals integral points for certain manifolds.

Paper finds at least 6 fixed points for a specific circle action on a 10D manifold.

In this paper, we study a circle action on a compact oriented manifold with a discrete fixed point set. The fixed point data consists of the weights of the $S^1$-representations at the fixed points. We prove various results and properties of the action, in terms of the fixed point data. We show that the manifold can be…

New proof for 6D symplectic manifold with 4 fixed points.

A pair of points in a riemannian manifold $M$ is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in $M$ are secure. A manifold is insecure if there exists an insecure point pair, and to…

The paper studies Morse flows on 3-manifold boundaries with fixed points.

Study fixed-point sets of $S^{1}$-actions on quaternionic manifolds.

Classifies multigraphs for torus actions on 6D manifolds with isolated fixed points.

Classifies circle actions on 6D manifolds with 4 fixed points.

The author proved that if the circle acts symplectically on a compact, connected symplectic manifold $M$ with three fixed points, then $M$ is equivariantly symplectomorphic to some standard action on $\mathbb{CP}^2$. In this paper, we extend the result to a circle action on an almost complex manifold; if the circle act…

Riemannian gradient descent escapes some spurious critical points on low-rank matrix manifold.

New method produces reflections with nonseparating fixed points.

We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case we prove a strengthened Price inequality. We employ these inequalit…

Paper extends previous result on hypersurfaces with degenerate light-like points.

Study a 10D symplectic manifold with 6 fixed points, linking to $G_2$ orbit.

Let $f:V^n\looparrowright M^m$ be a smooth generic immersion. Then the set of points, that have at least $k$ preimages is an image of a (non-generic) immersion. If the manifolds $V^n$ and $M^m$ are oriented and $m-n$ is even, then the manifold of $k$-fold points is also oriented. In this paper we compute the oriented b…

Estimates geodesics on surfaces without conjugate points.

The paper proves Morse estimates for translated points on unit tangent bundles.

Theorems on the existence of vector fields with given sets of Indexes of isolated Singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a two-dimensional manifold, an index of an isolated Singular point of the gradient field…

The minimal number of critical points is studied for smooth functions on closed manifolds.

We present a homogenization theorem for isotropically-distributed point defects, by considering a sequence of manifolds with increasingly dense point defects. The loci of the defects are chosen randomly according to a weighted Poisson point process, making it a continuous version of the first passage percolation model.…

Sampling random points can reveal submanifold topology.

Two-root Riemannian manifolds have no odd-dimensional examples.

Constructs stable maps from 3-manifolds to surfaces without cusps.

To each isolated critical point of a smooth function on a 3-manifold we put in correspondence a tree (graph without cycles). We will prove that functions are topologically equivalent in the neighborhoods of critical points if and only if the corresponding trees are isomorphic. A complete topological invariant of functi…

The paper proves critical point results for Frechet manifolds.

The triple point numbers and the triple point spectrum of a closed 3-manifold were defined in (R. Vigara, Representación de 3-variedades por esferas de Dehn rellenantes, PhD Thesis, UNED 2006). They are topological invariants that give a measure of the complexity of a 3-manifold using the number of triple points of min…

Estimates for solutions on manifolds under Ricci flow.

Study circle actions on 4-manifolds, deriving formulas and graphs.

We study fixed points of smooth torus actions on closed manifolds using fixed point formulas and equivariant elliptic genera. We also give applications to positively curved Riemannian manifolds with symmetry.

The study examines how shallow neural nets converge to training samples or manifold points during diffusion.

We introduce a new operation, double point surgery, on immersed surfaces in a 4-manifold, and use it to construct knotted configurations of surfaces in many 4-manifolds. Taking branched covers, we produce smoothly exotic actions of Z/m x Z/n on simply connected 4-manifolds with complicated fixed-point sets.

A manifold is locally \emph{$k$-fold symmetric}, if for any point and any $k$-dimensional vector subspace tangent to this point there exists a local isometry such that this point is a fixed point and the differential of the isometry restricted to that $k$-dimensional vector subspace is minus the identity. We show that …

Modern sample points in many applications no longer comprise real vectors in a real vector space but sample points of much more complex structures, which may be represented as points in a space with a certain underlying geometric structure, namely a manifold. Manifold learning is an emerging field for learning the unde…

The Brouwer fixed point theorem says that any continuous function from disc to itself has a fixed point. By using simple geometrical technique we have generalized the result in manifold and proved that any continuous function on the boundary of a bounded convex domain of a $2$-dimensional Riemannian manifold with a pol…

The study examines continuous mean curvature functions on manifolds without conjugate points.

Maps with boundary definite fold points restrict manifold structure.

Surfaces in 3-manifolds concentrate at curvature critical points.

Minimal surfaces' boundary points are always smooth.

The Poisson equation on manifolds plays an fundamental role in many applications. Recently, we proposed a novel numerical method called the Point Integral method (PIM) to solve the Poisson equations on manifolds from point clouds. In this paper, we prove the convergence of the point integral method for solving the Pois…

The study computes Bergman kernels and point process asymptotics on Kähler manifolds.