arXiv research

Study intrinsic volume forms on complex hypersurfaces.

Any Riemannian manifold has a canonical collection of valuations (finitely additive measures) attached to it, known as the intrinsic volumes or Lipschitz-Killing valuations. They date back to the remarkable discovery of H. Weyl that the coefficients of the tube volume polynomial are intrinsic invariants of the metric. …

For any closed smooth Riemannian manifold H. Weyl has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of th…

Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov-Hausdorf…

Weyl's intrinsic volumes converge to the Euler characteristic of the base manifold under certain metrics.

The paper proves stability of manifolds with boundary under volume and distance constraints.

Study shows convergence of volumes on manifolds with boundary under area constraints.

We explicitly compute the intrinsic volume of the set of real (and real symmetric) matrices of Frobenius norm one and given corank (the case of matrices with zero determinant as a special case). We give asymptotic formulas for our computations and we discuss several examples and applications.

The intrinsic entropy model accurately estimates stock market volatility.

Study on predicting sequences with Gaussian constraints, linking to intrinsic volumes and metric complexity.

On a sub-Riemannian manifold we define two type of Laplacians. The \emph{macroscopic Laplacian} $Δ_ω$, as the divergence of the horizontal gradient, once a volume $ω$ is fixed, and the \emph{microscopic Laplacian}, as the operator associated with a sequence of geodesic random walks. We consider a general class of rando…

We establish formulas that give the intrinsic volumes, or curvature measures, of sublevel sets of functions defined on Riemannian manifolds as integrals of functionals of the function and its derivatives. For instance, in the Euclidean case, if $f \in \mathcal{C}^3(\mathbb{R}^n, \mathbb{R})$ and 0 is a regular value of…

In this paper, we study the complete bounded $λ$-hypersurfaces in weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded $λ$-hypersurfaces with $|A|\leqα$ and get some applications of the volume comparison theorem. Secondly, we consider the relation amo…

Hadwiger's Theorem states that Euclidean-invariant convex-continuous valuations of definable sets are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable real-valued functions on n-dimensional Euclidean space. This generalizes intrinsic vol…

Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.

The paper proves convergence of metrics to a limit in a specific geometric context.

Study semicontinuity of capacity in non-smooth spaces using intrinsic flat convergence.

Study on volume growth of horospheres in specific Heintze groups.

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives …

In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces $Σ\subset\mathbb{R}^{n+1}$. We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on the norm of the second fundamental form. Furthermore, we prove an equivalence of pro…

A description of continuous rigid motion compatible Minkowski valuations is established. As an application, we present a Brunn-Minkowski type inequality for intrinsic volumes of these valuations.

We relate $L^p$ convergence of metric tensors or volume convergence to a given smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence for sequences of Riemannian manifolds. We present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general ev…

We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. …

Spherical Plateau problem studies minimal surfaces in quotients of spheres.

Researchers classify and decompose valuations on convex functions.

We study sequences of integral current spaces $(X_j,d_j,T_j)$ such that the integral current structure $T_j$ has weight $1$ and no boundary and, all $(X_j,d_j)$ are closed Alexandrov spaces with curvature uniformly bounded from below and diameter uniformly bounded from above. We prove that for such sequences either the…

In this note, we extend the Bridgeman-Kahn identity to all finite-volume orientable hyperbolic $n$-manifolds with totally geodesic boundary. In the compact case, Bridgeman and Kahn are able to express the manifold's volume as the sum of a function over only the orthospectrum. For manifolds with non-compact boundary, ou…

Weyl's tube formula holds for various cross-sections under symmetry conditions.

Study shows tori metrics converging to flat under specific conditions.

We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal h…

We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff sense to a countably $\mathcal{H}^m$ rectifiable metric space of the…

We consider the volume expansion of the Blaschke metric, which is a projectively invariant metric on a strictly convex domain in a locally flat projective manifold. When the boundary is even dimensional, we express the logarithmic coefficient L as the integral of affine invariants over the boundary. We also formulate a…

This paper constructs a function on Gromov-Hausdorff limits of 2-surfaces with curvature constraints.

The paper proves convex bodies are minimal fillings and have Lipschitz-volume rigidity.

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, whil…

Let $(M,I)$ be an almost complex 6-manifold. The obstruction to integrability of almost complex structure (so-called Nijenhuis tensor) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold is called nearly Kaehler if it adm…

This paper discuss an intrinsic relation among congruent relations \cite{CLPZ}, cyclotomic expansion and Volume Conjecture for $SU(n)$ invariants. Motivated by the congruent relations for $SU(n)$ invariants obtained in our previous work \cite{CLPZ}, we study certain limits of the $SU(n)$ invariants at various roots of …

We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself.…

The paper examines sequences of metric spaces converging to compact limits with specific properties.

We revisit and demonstrate the Epps effect using two well-known non-parametric covariance estimators; the Malliavin and Mancino (MM), and Hayashi and Yoshida (HY) estimators. We show the existence of the Epps effect in the top 10 stocks from the Johannesburg Stock Exchange (JSE) by various methods of aggregating Trade …

We study the classification of special almost hermitian manifolds in Gray and Hervella's type classes. We prove that the exterior derivatives of the symplectic form and the complex volume form contain all the information about the intrinsic torsion of the $\SUn(n)$-structure. Furthermore, we apply the obtained results …

3D spheres with certain properties approach the round sphere.

The study shows properties of stable anisotropic minimal hypersurfaces in 4D space.

Suppose a sequence $M_j$ of Alexandrov spaces collapses to a space $X$ with only weak singularities. Yamaguchi constructed a map $f_j:M_j\to X$ called an almost Lipschitz submersion for large $j$. We prove that if $M_j$ has a uniform positive lower bound for the volumes of spaces of directions, which is sufficiently la…

In this paper we address the relationship between Gromov-Hausdorff limits and intrinsic flat limits of complete Riemannian manifolds. In \cite{SormaniWenger2010, SormaniWenger2011}, Sormani-Wenger show that for a sequence of Riemannian manifolds with nonnegative Ricci curvature, a uniform upper bound on diameter, and n…

New proof shows affine manifolds with parallel volume are Riemannian-flat.

For negatively curved manifolds, a condition is found for intrinsic ultracontractivity of heat semigroups.

We relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular at…