arXiv research

Study on compact Kähler surfaces for sign-changing curvatures.

Estimates graph curvature and diameter using Laplacian eigenvalues.

Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.

We associate to any Riemannian symmetric space (of finite or infinite dimension) a L$^*$-algebra, under the assumption that the curvature operator has a fixed sign. L$^*$-algebras are Lie algebras with a pleasant Hilbert space structure. The L$^*$-algebra that we construct is a complete local isomorphism invariant and …

New Einstein metrics found on manifolds with opposite curvature signs.

A new algorithm solves signed Fréchet regression on manifolds with bounded curvature.

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 …

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…

This study examines how removing edges from complete graphs affects Ollivier Ricci curvature.

Ancient Ricci flows are identified without curvature sign condition.

For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.

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

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

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.

Researchers solve the negative Yamabe case for scalar curvature prescription.

The paper solves a problem related to curvature in complex geometry.

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

Study Euler characteristic of manifolds with almost nonnegative curvature operator, showing nonnegativity under certain conditions.

The study investigates deformations of swallowtails in 3D space, preserving curvature signs.

We obtain a vanishing theorem for the kernel of a Dirac operator on a Clifford module twisted by a sufficiently large power of a line bundle, whose curvature is non-degenerate at any point of the base manifold. In particular, if the base manifold is almost complex, we prove a vanishing theorem for the kernel of a $\spi…

We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yam…

We prove that the limit hypersurfaces of converging curvature flows are stable, if the initial velocity has a weak sign, and give a survey of the existence and regularity results.

New heat trace coefficients reveal curvature effects in polygonal domains.

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.

Paper solves Gauduchon scalar curvature problem on almost Hermitian manifolds.

New curvature K(x) measures manifold properties without integrals.

The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar curvature Riemannian metrics g on M. (To be absolutely precise, one only considers constant-scalar-curvature metrics which are Yamabe minimizers, but this does not affect the sign of th…

Classifies and constructs translators for curvature flows.

We study the geometry at infinity of expanding gradient Ricci solitons of dimension greater than two with finite asymptotic curvature ratio without curvature sign assumptions. We mainly prove that they have a cone structure at infinity.

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

We calculate the Riemann curvature tensor and sectional curvature for the Lie group of volume-preserving diffeomorphisms of the Klein bottle and projective plane. In particular, we investigate the sign of the sectional curvature, and find a possible disagreement with a theorem of Lukatskii. We suggest an amendment to t…

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…

Derives curvature conditions for spatial isotropy without field equations.

New cosmological models with changing curvature slices.

Ridgeless ReLU networks interpolate datasets and extrapolate based on curvature signs.

The study links Ricci curvature and convexity in complex tori.

Ricci curvature links volume convexity and minimal submanifolds.

Theory proves existence of hypersurfaces with prescribed curvature.

We show that there are high-dimensional smooth compact manifolds which admit pairs of Einstein metrics for which the scalar curvatures have opposite signs. These are counter-examples to a conjecture considered by Besse. The proof hinges on showing that the Barlow surface has small deformations with ample canonical line…

Study of Ricci flow convergence on surfaces with boundary.

Two spheres found with specific curvature constraints.

In the 3-dimensional Lorentz-Minkowski space we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also investigate the behavior of the Gaussian curvature near singular points o…

In a conformal class of metrics with positive Yamabe invariant, we derive a necessary and sufficient condition for the existence of metrics with positive Q curvature. The condition is conformally invariant. We also prove some inequalities between the Green's functions of the conformal Laplacian operator and the Paneitz…

Local solubility of Bao--Ratiu equations proven for surfaces with specific curvature conditions.

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…

In this paper we classify all non-Berwaldian Randers metrics of Douglas type arising from invariant hyper-Hermitian metrics on simply connected four-dimensional real Lie groups. Also, the formulas of the flag curvature are given and it is shown that, in some directions, the flag curvature of the Randers metrics and the…

We prove the existence of metrics with prescribed $Q$-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a pre…

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…