arXiv subjects

Thalia Jeffres

Publications and source records attributed to Thalia Jeffres.

4 recordsLinked to original sources

arXiv2026

Li-Yau and Harnack estimates for nonlinear parabolic equations

We establish global Harnack inequalities for positive viscosity solutions of several classes of nonlinear parabolic equations. The equations include positively homogeneous uniformly parabolic flows on Euclidean space and flat tori, normalized weighted $p$-Laplacian heat flows, and homogeneous isotropic diffusions on closed smooth metric measure spaces. For the normalized infinity-Laplacian heat flow, no curvature assumption is needed. For the ordinary weighted $p$-Laplacian, we obtain pressure estimates under nonnegative Bakry-Émery Ricci lower bounds. The proofs combine comparisons at two independent space-time points using the maximum principle for semicontinuous functions with weighted geodesic variations. The corresponding Li-Yau inequalities then follow in the first-order viscosity sense.

math.AP↗

arXiv2025

Gradient bounds for viscosity solutions to certain elliptic equations

Our principal object of study is the modulus of continuity of a periodic or uniformly vanishing function $ u: \mathbb{R} ^{n} \rightarrow \mathbb{R} $ which satisfies a degenerate elliptic equation $ F(x, u, \nabla u, D^{2} u) = 0 $ in the viscosity sense. The equations under consideration here have second-order terms of the form $ -{\rm Trace} \, (\mathcal{A} (\|\nabla u \|) \cdot D^{2} u) , $ where $ \mathcal{A} $ is an $ n\times n$ matrix which is symmetric and positive semi-definite. Following earlier work, \cite{Li21}, of the second author, which addressed the parabolic case, we identify a one-dimensional equation for which the modulus of continuity is a subsolution. In favorable cases, this one-dimensional operator can be used to derive a gradient bound on $u$ or to draw other conclusions about the nature of the solution.

math.AP↗

arXiv2021

Cohomogeneity one central Kähler metrics in dimension four

A Kähler metric is called central if the determinant of its Ricci endomorphism is constant. For the case in which this constant is zero, we study on $4$-manifolds the existence of complete metrics of this type which are cohomogeneity one for three unimodular $3$-dimensional Lie groups: $SU(2)$, the group of Euclidean plane motions $E(2)$ and a quotient by a discrete subgroup of the Heisenberg group $\mathrm{nil}_3$. We obtain a complete classification for $SU(2)$, and some existence results for the other two groups, in terms of specific solutions of an associated ODE system.

math.DG↗

arXiv2021

Conformal deformations of conic metrics to constant scalar curvature

We consider conformal deformations within a class of incomplete Riemannian metrics which generalize conic orbifold singularities by allowing both warping and any compact manifold (not just quotients of the sphere) to be the "link" of the singular set. Within this class of "conic metrics," we determine obstructions to the existence of conformal deformations to constant scalar curvature of any sign (positive, negative, or zero). For conic metrics with negative scalar curvature, we determine sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1; moreover, we show that this metric is unique within its conformal class of conic metrics. Our work is in dimensions three and higher.

math.DG↗