arXiv research

Introduces differential forms to study inequalities between eigenvalues.

New Poincaré inequality for differential forms on manifolds.

In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo, we obtain differential Harnack inequalities for the positive solutions to the Schödinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequali…

We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the $\varepsilon$-Ricci flow on a closed surface. These new Harnac…

Establishes Calderón-Zygmund inequalities on evolving Riemannian manifolds.

We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a concavity maximum principle argument to the corresponding level set f…

The paper proves inequalities for twisted differential forms on manifolds.

We prove a Poincare type inequality for differential forms on compact manifolds by means of a constructive 'globalization' of a local Poincare inequality on convex sets.

The paper proves inequalities on Riemannian manifolds using a test function method.

The paper extends inequalities to twisted differential forms on Kähler manifolds.

New proof of Gaffney's inequality for differential forms on manifolds with boundary.

We study a generalized Abreu Equation in $n$-dimensional polytopes and prove some differential inequalities for homogeneous toric bundles.

Poincar{é} and Sobolev inequalities for differential forms on Heisenberg balls, involving Rumin's differentials, are given. Furthermore, a global homotopy of Rumin's complex which improves differentiability of Rumin forms is provided on any bounded geometry contact manifold.

Paper proves inequalities for forms on sub-Riemannian manifolds.

Derives inequality for optimal transport on manifolds.

In [10], R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator. We show that this inequality holds under the weaker condition that M x R^2 has nonnegative isotropic curvature.

The paper studies boundedness of pseudo-differential operators on smooth manifolds.

Study improves Poincaré-Sobolev inequalities for differential forms.

Proves Kato inequalities for various conformal operators.

Data processing inequalities link Fisher information to local differential privacy constraints.

In this paper, we prove Poincaré and Sobolev inequalities for differential forms in $L^1(\mathbb R^n)$. The singular integral estimates that it is possible to use for $L^p$, $p>1$, are replaced here with inequalities which go back to Bourgain-Brezis.

We prove a version of differential Harnack inequality for a family of sub-elliptic diffusions on Sasakian manifolds under certain curvature conditions.

The paper proves nonexistence results for certain parabolic inequalities on Riemannian manifolds.

The paper characterizes curvature-dimension conditions and related inequalities on Riemannian manifolds.

Researchers prove inequalities for differential forms in Heisenberg groups, extending Euclidean results.

In this paper, we prove interior Poincar{é} and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike for L p , p > 1, the estimates are doomed to fail in top degree. The singular integ…

We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation $ω_t=Δω+aω\ln ω$ on closed manifolds. We also derive a new interpolated Harnack inequality for the equation $ω_t=Δω-ω\lnω+\varepsilon Rω$ on closed surfaces under the $\varepsilon$-Ricci flow. Finally we prove…

Proves spectral inequality and null-controllability for elliptic operators on closed manifolds.

We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold $(M,g)$ and the $L_{q,p}$-cohomology of that manifold. The $L_{q,p}$-cohomology of $(M,g)$ is defined to be the quotient of the space of closed differential forms in $L^p(M)$ modulo the exact forms which are exterior diff…

The paper extends Liouville theorems to sub-Riemannian manifolds.

The paper studies eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.

Study on biharmonic Steklov problem on differential forms.

Derives formulas for differential forms on weighted manifolds.

Paper introduces magnetic Steklov operator on differential forms and its properties.

We prove a refined Kato inequality for closed and coclosed differential $(p,q)$ forms on a Kahler manifold.

New proof of Willmore inequality using geometric divergence inequality.

We prove that on a Riemannian manifold, a smooth differential form has a primitive with a given (functional) upper bound provided the necessary weighted isoperimetric inequalities implied by Stokes are satisfied. We apply this to prove a comparison predicted by Gromov between the cofilling function and the filling area…

Study on biharmonic Steklov problems with Neumann boundary conditions and eigenvalue estimates.

Overview of recent results on isoperimetric inequalities on manifolds with Ricci lower bounds.

For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of varifold theory in the study of diffused surfaces.

In this paper we are concerned with a class of elliptic differential inequalities with a potential in bounded domains both of $\mathbf{R}^m$ and of Riemannian manifolds. In particular, we investigate the effect of the behavior of the potential at the boundary of the domain on nonexistence of nonnegative solutions.

In this paper we are concerned with a class of elliptic differential inequalities with a potential both on $\erre^m$ and on Riemannian manifolds. In particular, we investigate the effect of the geometry of the underlying manifold and of the behavior of the potential at infinity on nonexistence of nonnegative solutions.

The paper explores dualities in differential equations and their applications in Riemannian geometry.

In recent papers Wu-Yau, Tosatti-Yang and Diverio-Trapani, used some natural differential inequalities for compact Kähler manifolds with quasi negative holomorphic sectional curvature to derive positivity of the canonical bundle. In this note we study the equality case of these inequalities.

We give a proof to the Li-Yau-Hamilton type inequality claimed by Perelman on the fundamental solution to the conjugate heat equation. The rest of the paper is devoted to improving the known differential inequalities of Li-Yau-Hamilton type via monotonicity formulae.

In this paper we state and prove Morse type inequalities for Morse functions as well as for closed differential 1-forms. These inequalities involve delocalized Betti numbers. As an immediate consequence, we prove the vanishing of delocalized Betti numbers of manifolds fibering over the circle.

The paper extends a Harnack inequality to noncompact evolving hypersurfaces.

The paper introduces Morse theory for Lie groupoids and proves inequalities.