arXiv research

New curvature for weighted Sasaki sphere found.

Counting HCMU sphere components using weighted trees.

Solves weighted bi-colored plane tree enumeration and applies to geometric problems.

Proves existence of curves with constant curvature in a sphere.

Any Sasakian structure can be closely mimicked by embeddings into weighted spheres.

The paper extends geometric inequalities for nearly spherical sets in various space forms.

The paper proves geometric inequalities in sphere using locally constrained flows.

Spheres minimize weighted curvature on spheres.

Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries. Results on weighted best and random approximation and the new approach to floati…

The paper proves new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.

Constructs geometric models for moduli spaces of Higgs bundles over Riemann sphere.

We prove two weighted geometric inequalities that hold for strictly mean convex and star-shaped hypersurfaces in Euclidean space. The first one involves the weighted area and the area of the hypersurface and also the volume of the region enclosed by the hypersurface. The second one involves the total weighted mean curv…

We obtain upper estimates for the bottom (that is, greatest lower bound) of the essential spectrum of weighted Laplacian operator of a weighted manifold under assumptions of the volume growth of their geodesic balls and spheres. Furthermore, we find examples where the equality occurs in the estimates obtained. As a con…

The paper analyzes various normalization methods in deep learning.

The paper extends rigidity results for $λ$-self-expanders to hyperplanes, spheres, and cylinders.

Study Cremona transformations in weighted projective planes to find rational cuspidal curves and Zariski pairs.

New summary measures reveal geometric structure in weighted measures on manifolds.

The paper studies variations of weighted curvature on submanifolds.

In this paper we discuss the stability of geodesic spheres in $\mathbb{S}^{n+1}$ under constrained curvature flows. We prove that under some standard assumptions on the speed and weight functions, the spheres are stable under perturbations that preserve a volume type quantity. This extends results by Escher and Simonet…

We determine the curvature equations of natural metrics on tangent bundles and radius r tangent sphere bundles S_rM of a Riemannian manifold M. A family of positive scalar curvature metrics on S_rM is found, for any M with bounded sectional curvature and any chosen constant r.

Classifies smooth metric measure spaces with two weighted Einstein representatives.

New surgery obstructions found in simple character varieties.

Study preserves metrics with positive Bakry-Émry Ricci curvature via surgery.

We prove that, for a Finsler space, if the weighted Ricci curvature is bounded below by a positive number and the diam attains its maximal value, then it is isometric to a standard Finsler sphere. As an application, we show that the first eigenvalue of the Finsler-Laplacian attains its lower bound if and only if the Fi…

We consider a geometrically finite discrete group of conformal transformations of the sphere. Further we consider distributions which are supported on the limit set and are invariant with conformal weight. We estimate their regularity in terms of the conformal weight, the Hausdorff dimension of the limit set, and the m…

We give a construction of Kirby weight systems associated to sl(2) and valued into the finite field Z/pZ. We show that it is possible to apply this sequence of weight systems on the universal invariant of framed link. We also show that the corresponding sequence admits a Fermat limit, which defines an asymptotic ration…

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modifie…

We prove a Theorem on homotheties between two given tangent sphere bundles $S_rM$ of a Riemannian manifold $M,g$ of $\dim\geq 3$, assuming different variable radius functions $r$ and weighted Sasaki metrics induced by the conformal class of $g$. New examples are shown of manifolds with constant positive or with constan…

Proves existence of special 2-spheres in curved 3-spaces.

Study on rank 2 Higgs bundles on 5-punctured sphere, proving $P=W$ conjecture in lowest degree.

A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the weighted number of convex tilings with a given number of tiles. To prove these for…

Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.

Flat semigroups can represent normal weighted homogeneous surface singularities.

The paper proves rigidity results for self-shrinkers and surfaces with parallel weighted mean curvature.

We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variation…

We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd …

We study a variation of Bagchi and Datta's $σ$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated…

Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical semigroups. More precisely, we prove that the strongly flat semigroups, which satisfy the…

Compact character varieties of punctured spheres are proven.

Study on Higgs bundles and hyperpolygon spaces using Hitchin metrics.

We establish a one-to-one correspondence between Finsler structures on the $2$-sphere with constant curvature $1$ and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics are closed on the other hand. As a…

Natural metric structures on the tangent bundle and tangent sphere bundles $S_rM$ of a Riemannian manifold $M$ with radius function $r$ enclose many important unsolved problems. Admitting metric connections on $M$ with torsion, we deduce the equations of induced metric connections on those bundles. Then the equations o…

Fermat constants fail to fully identify Clairaut constants for certain geodesics on a surface of revolution.

We attempt to define a new invariant I of (almost) Calabi-Yau 3-folds M, by counting special Lagrangian rational homology 3-spheres N in M in each 3-homology class, with a certain weight w(N) depending on the topology of N. This is motivated by the Gromov-Witten invariants of a symplectic manifold, which count the J-ho…

Study local moduli of Sasaki-Einstein metrics on specific polynomial links.

Proves representability of complex semigroup systems.

Classifies surfaces of section for Seifert fibrations.

Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.