arXiv research

Study on compact Kähler surfaces for sign-changing curvatures.

We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry-Emery will transform under time change. Similarly, for metric measure spaces we study how the curvature-dimension c…

Lectures on mean curvature flow and its related equations.

Study nondegenerate singularities in mean curvature flow.

In this paper, we consider the problem of prescribing scalar curvature on n-sphere. Assume that the candidate curvature function $f$, which is allowed to change sign, satisfies some kind of Morse index or symmetry condition. By studying the well-known scalar curvature flow, we are able to prove that the flow converges …

The paper proves properties of strain tensors on surfaces with changing Gauss curvature.

The present paper is a continuation of a foregoing paper [Tensor, N. S., 69 (2008), 155-178]. The main aim is to establish \emph{an intrinsic investigation} of the conformal change of the most important special Finsler spaces, namely, $C^{h}$-recurrent, $C^{v}$-recurrent, $C^{0}$-recurrent, $C_{2}$-like, quasi-$C$-redu…

Study proves a new flow method for mean curvature with volume change analysis.

This paper focuses on the problem of prescribing mean curvature on the unit ball. Assume that $f$, which is allowed to change sign, satisfies Morse index counting or certain kind of symmetry condition. By using a negative gradient flow method, we then prove that $f$ can be realized as the boundary mean curvature of som…

The paper solves curvature prescription problems on balls and disks.

The study explores spacetimes with changing spatial curvature, leading to topological transitions.

New cosmological models with changing curvature slices.

Study Ricci curvature of homogeneous Finsler spaces with specific metrics.

Study identifies obstructions for solving a 4th-order boundary problem.

Study on curvature flow in 4D ball, proving existence and convergence.

Study variations of Riemannian submersions to maintain geodesic fibers and positive curvatures.

In this paper, we prove that every m-th root metric with isotropic mean Berwald curvature reduces to a weakly Berwald metric. Then we show that an m-th root metric with isotropic mean Landsberg curvature is a weakly Landsberg metric. We find necessary and sufficient condition under which conformal $β$-change of an m-th…

In this paper we classify complete surfaces of constant mean curvature whose Gaussian curvature does not change sign in a simply connected homogeneous manifold with a 4-dimensional isometry group.

It is well-known that space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski 3-space R^3_1 have singularities in general. They are both characterized as zero mean curvature surfaces. We are interested in the case where the singular set consists of a light-like line, since this case has not been…

Researchers solve the negative Yamabe case for scalar curvature prescription.

The paper generalizes CR invariants using renormalized characteristic forms.

This note corrects a mistake in the original book in the evolution equations of total curvature for the curve-shrinking flow in an ambient Ricci Flow. The resulting upper bound for the evolution of total curvature is an exponential bound in time. The change involves the multiplicative constant. Here we show that it dep…

Study finds least-energy nodal solutions to the Yamabe problem on manifolds with boundary.

The main purpose of this short note is to point out that the negative gradient flow for the prescribed $\mathbf Q$-curvature problem on $S^n$ can be extended to handle the case that the $\mathbf Q$-curvature candidate $f$ may change signs.

Study on pseudo-Einstein 3-manifolds, calculating determinant changes under conformal transformations.

We compute renormalized curvature integrals on Poincaré-Einstein manifolds.

We propose a natural definition of the weighted $σ_k$-curvature for a manifold with density; i.e.\ a triple $(M^n,g,e^{-φ}\mathrm{dvol})$. This definition is intended to capture the key properties of the $σ_k$-curvatures in conformal geometry with the role of pointwise conformal changes of the metric replaced by pointw…

We consider flows, called $W^{\rm u}$ flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of $W^{\rm u}$ flows and we show that $W^{\rm u}$ flows have purely absolutely continuous spectrum in the orthocom…

Mapping complex input data into suitable lower dimensional manifolds is a common procedure in machine learning. This step is beneficial mainly for two reasons: (1) it reduces the data dimensionality and (2) it provides a new data representation possibly characterised by convenient geometric properties. Euclidean spaces…

This is not in any way meant to be a complete survey on positive curvature. Rather it is a short essay on the fascinating changes in the landscape surrounding positive curvature. In particular, details and many results and references are not included, and things are not presented in chronological order.

Study optimal partition problem for Q-curvature equations on Einstein manifolds.

Proves existence of multi-phase flows from arbitrary initial data.

Singular Yamabe problems involve changing sign solutions with interesting geometric properties.

In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with non-vanishing mean curvatur…

Study describes how conformal metrics behave as Q-curvature changes, forming spherical bubbles.

Consider an orientable compact surface in three dimensional Euclidean space with minimum total absolute curvature. If the Gaussian curvature changes sign to finite order and satisfies a nondegeneracy condition along closed asymptotic curves, we show that any other isometric surface differs by at most a Euclidean motion…

Theory proves existence of hypersurfaces with prescribed curvature.

The paper studies curvature changes on manifolds with boundary.

In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the…

We study topology change in (2+1)D gravity coupling with non-Abelian SO(2,1) Higgs field from the point of view of Morse theory. It is shown that the Higgs potential can be identified as a Morse function. The critical points of the latter (i.e. loci of change of the spacetime topology) coincide with zeros of the Higgs …

In the present paper, we find the conditions to characterize projective change between two $(α, β)$-metrics, F = $α+ εβ+ k\frac{β^2}α$ ($ε$ and k $\neq$ 0 are constants) and a Matsumoto metric $\bar{F}=\frac{\barα^2}{\barα -\barβ}$ on a manifold with dimension $n \geq 3$ where $α$ and $\barα$ are two Riemannian metrics…

In this paper we show that an immersed nontrivial translating soliton for mean curvature flow in $\mathbb{R}^{n+1}$($n=2,3)$ is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most quadratic growth. For an embedded translating soliton $Σ$ with nonnegative scalar curva…

Two spheres found with specific curvature constraints.

New framework to understand and exploit curvature in deep learning loss landscapes.

Uniform curvature bounds for regularized metrics with bounds on Ricci tensor and injectivity radius.

Study on prescribing curvature on specific manifolds with negative Gauduchon degree.

This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for $n \geq 3$. Given a rotationally symmetric function $H:\partial B^{n}\rightarrow R$, in this work, we will prove that if $H'(r)$ changes signs where $H>0$ and $H(r)$ also satisfies a flatness con…

Study proves curvature prescription on spheres for k ≥ n/2.