arXiv research

Study area and coarea formulas for graphs and submanifolds in Carnot groups.

The coarea formula is proven for Heisenberg group maps, addressing open questions.

We present the area and coarea formulas for Lipschitz maps, valid for general volume densities. As applications, we give a short, "euclidean" proof of the anisotropic Sobolev inequality and describe an anisotropic tube formula for hypersurfaces in $mathbb{R}^n$. A discussion about the first variation of the anisotropic…

New geometric measure simplifies complex analysis.

Proves inequality for 1-dimensional cycles.

We study a generalized functional related to the pullback metrics (3). We derive the first variation formula which yield stationary maps. We introduce the stress-energy tensor which is naturally linked to conservation law and yield monotonicity formula via the coarea formula and comparison theorem in Riemannian geometr…

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measu…

We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in $\mathbb{R}^2$. These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they e…

We generalize optimal inequalities of C. Loewner and M. Gromov, by proving lower bounds for the total volume in terms of the homotopy systole and the stable systole. Our main tool is the construction of an area-decreasing map to the Jacobi torus, streamlining and generalizing the construction of the first author in col…

Lipschitz maps on metric surfaces are rigid if they preserve area.

Paper proves method for calculating NML code length works for continuous models.

Derives Weyl law for volume spectrum using parametric inequalities.

Proves the Weyl law for 1-cycles in manifolds.

New geometric invariant limits the number of semi-arithmetic groups.

A variant of Gromov's H{ö}lder-equivalence problem, motivated by a pinching problem in Riemannian geometry, is discussed. A partial result is given. The main tool is a general coarea inequality satisfied by packing energies of maps.

Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2-complexes. Our inequality improves the constant in M. Gromov's inequ…

The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration by parts identities for certain compositions with smooth functions. In this class the idea of ze…

Proves a generalized isoperimetric inequality for spheres in dimensions 4 and above.

In this paper we deal with singular varieties of bounded mean curvature in the viscosity sense. They contain all varifolds of bounded generalized mean curvature. In the first part we investigate the second-order properties of these varieties, obtaining results that are new also in the varifold's setting. In particular …

In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with - or without - boundary $\partial S$; S and $\partial S$ are endowed with their homo…

Study on convergence of exponential probability measures with applications to maximum entropy models and SGLD.

Given a hyperelliptic Klein surface, we construct companion Klein bottles, extending our technique of companion tori already exploited by the authors in the genus 2 case. Bavard's short loops on such companion surfaces are studied in relation to the original surface so to improve a systolic inequality of Gromov's. A ba…

In 1972, Marcel Berger defined a metric invariant that captures the `size' of k-dimensional homology of a Riemannian manifold. This invariant came to be called the k-dimensional SYSTOLE. He asked if the systoles can be constrained by the volume, in the spirit of the 1949 theorem of C. Loewner. We construct metrics, ins…

Paper develops methods for analyzing forms with synchronized singularities.

We prove a new version of isoperimetric inequality: Given a positive real $m$, a Banach space $B$, a closed subset $Y$ of metric space $X$ and a continuous map $f:Y \rightarrow B$ with $f(Y)$ compact $\inf_FHC_{m+1}(F(X))\leq c(m)HC_m(f(Y))^{\frac{m+1}{m}},$ where $HC_m$ denotes the $m$-dimensional Hausdorff content,…

Proves a general connected sum formula for families Seiberg-Witten invariants.

Derives an integral formula for G2-structures.

Derives integral formulae on weighted manifolds.

Paper derives trace formula for magnetic Laplacian at zero energy.

The Gauss formula is extended to various Laplacians on submanifolds.

The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based…

We prove two tropical gluing formulae for Gromov-Witten invariants of exploded manifolds, useful for calculating Gromov-Witten invariants of a symplectic manifold using a normal-crossing degeneration. The first formula generalizes the symplectic-sum formula for Gromov-Witten invariants. The second formula is stronger, …

Note on new cancellation formulas for manifolds.

The main result of the present paper is a coincidence formula for foliated manifolds. To prove this we establish Kuenneth formula, Poincare duality and intersection product in the context of tangential de Rham cohomology and homology of tangential currents. We apply the formula to get a dynamical Lefschetz formula for …

Formula calculates volume of two-bridge knots.

Introduces a universal Bochner formula for scalar curvature.

Formula connects surgeries to Seiberg-Witten invariants.

It has been shown that the Alvarez-Gaum$\mathrm{\acute{e}}$-Witten miraculous anomaly cancellation formula in type IIB superstring theory and its various generalizations can be derived from modularity of certain characteristic forms. In this paper, we show that the Green-Schwarz formula and the Schwarz-Witten formula i…

Proves a formula for a special invariant of 4-manifolds.

Proves a special case of the Gaussian kinematic formula using large sphere limits.

New Crofton formulae derived from existing ones.

Formulae for non-symmetric connections derived from covariant derivatives.

Formula connects curvature to volume in special geometric spaces.

Kenmotsu's formula describes surfaces in Euclidean 3-space by their mean curvature functions and Gauss maps. In Lorentzian 3-space, Akutagawa-Nishikawa's formula and Magid's formula are Kenmotsu-type formulas for spacelike surfaces and for timelike surfaces, respectively. We apply them to a few problems concerning rota…

The paper proves T-duality and Hori formulae for winding loop spaces.

New formulas for measuring geometric properties of definable sets.

Guillemin trace formula adapted for group actions.

Unified entropy formula for real, complex, and quaternionic DLNs.