Searcharxiv⌕ Search

arXiv subjects

Ramesh Mete

Publications and source records attributed to Ramesh Mete.

6 recordsLinked to original sources
arXiv2026

A note on the positivity of $Q$-curvature via the continuity method

Suppose $(M, g)$ is a smooth, closed, $5$-dimensional Riemannian manifold with positive Yamabe invariant $Y(M, [g]) > 0$ and positive Yamabe-type $Q$-curvature invariant $Y_{4}^{\ast}(M, [g]) > 0$. Using the continuity method, we prove the existence of a metric in the conformal class $[g]$ with positive scalar curvature and positive $Q$-curvature, assuming an additional condition on an ``initial metric" in the class $[g]$.

math.DG↗
arXiv2026

On the rigidity of generalized $m$-quasi-Einstein manifolds of Yamabe-type

Motivated by the concept of almost Yamabe solitons, a special class of generalized $m$-quasi-Einstein manifolds is investigated in this paper. We refer to these Riemannian manifolds as generalized $m$-quasi-Einstein manifolds of Yamabe-type. We study the rigidity properties for the potential (or defining) vector field associated to these manifolds in both the compact and non-compact settings. We show that under certain natural assumptions the potential vector field either vanishes identically or becomes a non-trivial Killing vector field.

math.DG↗
arXiv2026

On the Killing property of the defining vector field for an almost Yamabe soliton

In this paper, we first investigate almost Yamabe solitons on compact Riemannian manifolds without boundary of dimension greater than or equal to two. We provide some sufficient conditions for which the defining conformal vector field associated to a compact almost Yamabe soliton is a Killing vector field. We then study almost Yamabe solitons on complete, non-compact Riemannian manifolds. We prove the Killing property of the defining conformal vector field associated to a complete, non-compact almost Yamabe soliton under certain conditions when the dimension is strictly greater than two.

math.DG↗
arXiv2024

Singularity formation in co-dimension one of the dHYM cotangent flow on blow up of $\mathbb{C}\mathbb{P}^{3}$ at a point

The existence and uniqueness of canonical singular solutions of the J-equation and the deformed Hermitian Yang Mills (dHYM) equation was proved in \cite{DMS24} on compact K\"{a}hler surfaces. In this paper, we study the singularity formation of the dHYM cotangent flow on the one-point blow up of $\mathbb{C}\mathbb{P}^3$ using Calabi ansatz. In particular, we provide an explicit example where the flow develops a singularity along the exceptional divisor. Moreover, the limit satisfies corresponding singular dHYM equation in the sense of \cite{DMS24} and provides some evidence for Conjecture $1.12$ in \cite{DMS24} on this three-dimensional manifold with symmetry.

math.DG↗
arXiv2023

Minimal slopes and bubbling for complex Hessian equations

The existence of smooth solutions to a broad class of complex Hessian equations is related to nonlinear Nakai type criteria on intersection numbers on Kahler manifolds. Such a Nakai criteria can be interpreted as a slope stability condition analogous to the slope stability for Hermitian vector bundles over Kahler manifolds. In the present work, we initiate a program to find canonical solutions to such equations in the unstable case when the Nakai criteria fails. Conjecturally such solutions should arise as limits of natural parabolic flows and should be minimisers of the corresponding moment-map energy functionals. We implement our approach for the J-equation and the deformed Hermitian Yang-Mills equation on surfaces and some examples with symmetry. We prove that there always exist unique canonical solutions to these two equations on Kahler surfaces in the unstable cases. Such canonical solutions with singularities are also shown to be the limits of the corresponding J-flow and the cotangent flow on certain projective bundles. We further present the bubbling phenomena for the J-equation by constructing minimizing sequences of the moment-map energy functionals, whose Gromov-Hausdorff limits are singular algebraic spaces.

math.DG↗