arXiv research

We give lower bounds for the first Dirichilet eigenvalues for domains in submanifolds with locally bounded mean curvatures. These bounds depend on the injectivity radius, sectional curvature (upperbound) of the ambient space and on the mean curvature of the submanifold. For submanifolds fo Hadamard manifolds these lowe…

Continuous curve evolution depends on initial shape on sphere.

New theorem shows curvature concentration depends linearly on volume ratio.

We prove that codimension two surfaces satisfying a nonlinear curvature condition depending on normal curvature are smoothly deformed by mean curvature flow to round points.

Study heat flow on changing surfaces, proving existence and uniqueness.

We prove new lower bounds for the first eigenvalue of the Dirac operator on compact manifolds whose Weyl tensor or curvature tensor, respectively, is divergence free. In the special case of Einstein manifolds, we obtain estimates depending on the Weyl tensor.

The paper proves finite Urysohn 1-width for 4 and 5 manifolds with positive biRicci curvature.

New equations reveal how cylinder power in progressive lenses depends on geodesic curvature.

In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary…

Study Wintgen ideal submanifolds in curved spaces with specific curvature conditions.

Local estimates of the maximal curvatures of admissible spacelike hypersurfaces in de Sitter space for k-symmetric curvature functions are obtained. They depend on interior and boundary data.

In this paper we present several curvature estimates for solutions of the Ricci flow which depend on smallness of certain local integrals of the norm of the Riemann curvature tensor.

We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with $|Ric|\leq K$ must satisfy $|Rm|\…

Localized curvature bounds ensure harmonic maps are constant.

Gradient and eigenvalue estimates for Kähler manifolds' canonical bundle.

Study heat kernel on quaternionic contact manifolds, finding linear dependence of coefficients on curvature.

We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.

This paper is the second part of a study of the quantum free particle on spherical and hyperbolic spaces by making use of a curvature-dependent formalism. Here we study the analogues, on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ ($κ>0$) and $H_\k^3$ ($κ<0$), to the standard {\itshape spherical wav…

We prove that, given an acausal curve $Γ$ in the boundary at infinity of $AdS_{3}$ which is the graph of a quasi-symmetric homeomorphism $φ$, there exists a unique foliation of its domain of dependence $D(Γ)$ by constant mean curvature surfaces with bounded second fundamental form. Moreover, these surfaces provide a fa…

Assuming a lower bound on the Ricci curvature of a complete Riemannian manifold, for $p< 1/2$ we show the existence of bounds on the local $L^p$ norm of the Ricci curvature that depend only on the dimension and which improve with volume collapse.

Develops local curvature estimates for mean curvature flow.

This paper concerns the evolution of a closed convex hypersurface in ${\mathbb{R}}^{n+1}$, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow e…

We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal (most negative) eigenvalue of the curvature operator at any point in the Teichmülle…

Sharp Sobolev and Michael-Simon inequalities on curved manifolds.

Proves existence and uniqueness of mean curvature flow.

In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2, then there exists a positive constant $δ$ depending on the ratio such that $\cosα\ge…

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of the nor…

Upper bound for total mean curvature of spin fill-ins is proven.

Integral of scalar curvature over a manifold is bounded by a constant depending on dimension and curvature threshold.

We present a unified derivation of covariant time derivatives, which transform as tensors under a time-dependent coordinate change. Such derivatives are essential for formulating physical laws in a frame-independent manner. Three specific derivatives are described: convective, corotational, and directional. The covaria…

We prove existence of compact spacelike hypersurfaces with prescribed k - curvature in de Sitter space, where the prescription function depends on both space and the tilt function.

Study p-Willmore disks with boundary energies, finding equilibrium configurations.

Study shows unique tangent flow for Lagrangian surfaces with bounded mean curvature.

Extends diameter bounds for submanifolds with boundary and minor curvature restrictions.

We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As applications, we obtain some curvature estimates of the Ricci shrinkers depending only on the non-collapsing constant.

Study submanifolds in spheres with Ricci curvature bounds.

Study spherical curves with curvature dependent on distance to a great circle.

We study the motion of smooth, strictly convex bodies in $\mathbb{R}^n$ expanding in the direction of their normal vector field with speed depending on Gauss curvature and support function.

Numerical simulations show stability of Type-II singularities in noncompact hypersurfaces.

We define a hybrid between Ollvier and Bakry Emery curvature on graphs with dependence on a variable neighborhood. The hexagonal lattice is non-negatively curved under this new curvature notion. Bonnet-Myers diameter bounds and Lichnerowicz eigenvalue estimates follow from the standard arguments. We prove gradient esti…

The total curvature of complex hypersurfaces in $\bC^{n+1}$ and its variation in families appear to depend not only on singularities but also on the behaviour in the neighbourhood of infinity. We find the asymptotic loss of total curvature towards infinity and we express the total curvature and the Gauss-Bonnet defect …

The main point of this paper is that, under suitable conditions on the mean curvature and the Ricci curvature of the ambient space, we can extend Choi-Schoen's Compactness Theorem to compact embedded minimal surfaces to simple immersed compact H-surfaces in a Riemannian manifold with positive Ricci curvature (the mean …

We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we int…

In this paper, we the improve the bound for the moment map derivative proved by Donaldson in his recent proof of the Hilbert-Mumford stability of complex manifolds with constant scalar curvature. The proof depends on the identification of Donaldson's symplectic form with the curvature of a certain Deligne pairing.

Riemannian manifolds with bounded Ricci curvature have finite Uryson width.

New method for constructing contact Lie systems on various spaces.

Lower bound found for Kähler manifold eigenvalues.

The Ricci flow preserves product structures with instantaneous curvature bounds.