arXiv research

Study simplicial volume for fixed fundamental groups, finding gaps.

Paper improves volume gap between minimal submanifolds and unit spheres.

Volume gaps for minimal submanifolds in spheres are proven.

New theorem shows curvature concentration depends linearly on volume ratio.

In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectra…

The paper validates a classifier for identifying intraday regime shifts in MNQ futures.

In this paper, we show that any ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean space for all time. This is a generalization of Anderson's result for Ricci-flat manifolds. As a corollary, a gap theorem …

The study examines hyperbolic 3-manifolds with uniform spectral gaps for coclosed 1-forms.

The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.

Torus covers have controlled volume and diameter under curvature and diameter bounds.

New volume functions for random hyperbolic surfaces link to spectral gaps.

Flat space for manifolds with tiny curvature.

The paper calculates mass and volume of Einstein metrics in four dimensions.

The paper studies 4D Ricci flow manifolds with curvature constraints.

In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature est…

Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for Ricci-flat manifolds, removing the volume growth assumptions in known results.

Study shows eigenvalue of Hodge Laplacian on coexact 1-forms in hyperbolic 3-manifolds is related to isoperimetric ratio.

To reduce the label complexity in Agnostic Active Learning (A^2 algorithm), volume-splitting splits the hypothesis edges to reduce the Vapnik-Chervonenkis (VC) dimension in version space. However, the effectiveness of volume-splitting critically depends on the initial hypothesis and this problem is also known as target…

Calibrations help estimate volumes on odd spheres without gaps.

On a compact $n$-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature o…

The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.

We study $λ$-hypersurfaces that are critical points of a Gaussian weighted area functional $\int_Σ e^{-\frac{|x|^2}{4}}dA$ for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems for complete $λ$-hypersurfaces in terms of the norm of the second fundamental form $|A|$. Sec…

Let $M$ be a connected, noncompact, complete Riemannian manifold, consider the operator $L=\DD +\nn V$ for some $V\in C^2(M)$ with $\exp[V]$ integrable w.r.t. the Riemannian volume element. This paper studies the existence of the spectral gap of $L$. As a consequence of the main result, let $\rr$ be the distance functi…

A new method for SSMF improves upon existing algorithms.

Geodesic spheres in certain symmetric spaces are quantitatively stable under small perturbations.

We have discovered a "little" gap in our proof of the sharp conjecture that in $\mathbb{R}^n$ with volume and perimeter densities $r^m$ and $r^k$, balls about the origin are uniquely isoperimetric if $0 < m \leq k - k/(n+k-1)$, that is, if they are stable (and $m > 0$). The implicit unjustified assumption is that the g…

Establishes refined singularity estimate for nonnegative n-superharmonic functions in locally conformally flat manifolds.

New framework uses trading volume instead of volatility for stock pricing.

We provide a proof for an inequality between volume and L2-Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau's theory of L2-Betti numbers of R-simplicial complexes, and other …

Optimizes dimension estimate for holomorphic functions on Kähler manifolds.

Study on positive scalar curvature and its impact on Ricci limit spaces.

New framework explains normalizing flows' power and limitations.

Paper studies a new curvature system and proves rigidity and gap theorems.

In this short note, using Günther's volume comparison theorem and Yokota's gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton $(M^{n},g,f)$ with sectional curvature $K(g)<A$ and ${\rm Vol}_{f}(M)\geq v$ for some uniform constant $A,v$, there exists…

We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently Gromov-Hausdorff close to rays, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spect…

Arithmetic spaces simplified to simplicial complexes.

Randomly glued tetrahedra form connected 3-manifolds with a single boundary.

The importance of considering the volumes to analyze stock prices movements can be considered as a well-accepted practice in the financial area. However, when we look at the scientific production in this field, we still cannot find a unified model that includes volume and price variations for stock assessment purposes.…

The study classifies complete self-shrinkers in Euclidean space.

We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric construction related to the discrete isoperimetric problem on the surface of a cone. In…

On curved spaces, viscous fluids reach equilibrium quickly.

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently construct…

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: 1) Geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results); 2) Volume estimates of metric balls; 3) Gradient bounds…

Study noncollapsed F-limit metric solitons, proving properties similar to smooth Ricci shrinkers.

The study proves a gap theorem for CAT(0) spaces with a constant below 1/(6√π).

Since $n$-dimensional $λ$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of complete $λ$-hypersurfaces. We give a gap theorem of complete $λ$-hypersurfaces with po…

We generalize a classification result for self-shrinkers of the mean curvature flow with nonnegative mean curvature, which was obtained by T. Colding and W. Minicozzi, replacing the assumption on polynomial volume growth with a weighted $L^2$ condition on the norm of the second fundamental form. Our approach adopt the …

Improved homological dimension for certain subgroups in Lie groups.