arXiv subjects

Chao-Ming Lin

Publications and source records attributed to Chao-Ming Lin.

5 recordsLinked to original sources

arXiv2026

Lower Bound for Weighted Intermediate Ricci Curvature and Tensorial Entropy Convexity

We introduce a weighted version of intermediate Ricci curvature and establish several equivalent characterizations of its lower bound. As an application, we generalize the results of Aishwarya--Rotem--Shenfeld [arXiv:2509.23399v1] by deriving intrinsic-dimensional evolution variational inequalities and the corresponding Wasserstein contraction estimates for the heat flow. We also compare our characterization with that of Ketterer--Mondino [arXiv:1610.03339v3] via lower-dimensional optimal transport.

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arXiv2023

On the Solvability of General Inverse $\sigma_k$ Equations

We prove that if there exists a $C$-subsolution to a constant coefficients strictly $\Upsilon$-stable general inverse $\sigma_k$ equation, then there exists a unique solution. As a consequence, this result covers all the analytical results of the classical strictly $\Upsilon$-stable general inverse $\sigma_k$ equations, for example, the complex Monge--Amp\`ere equation, the complex Hessian equation, the J-equation, the deformed Hermitian--Yang--Mills equation, etc. Hence, we confirm an analytical conjecture by Collins--Jacob--Yau [arXiv:1508.01934] of the solvability of the deformed Hermitian--Yang--Mills equation. Their conjecture states that the existence of a $C$-subsolution to a supercritical phase deformed Hermitian--Yang--Mills equation gives the solvability.

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arXiv2022

On the Convexity of General Inverse $\sigma_k$ Equations

We prove that if a level set of a degree $n$ general inverse $\sigma_k$ equation $f(\lambda_1, \cdots, \lambda_n) = \lambda_1 \cdots \lambda_n - \sum_{k = 0}^{n-1} c_k \sigma_k(\lambda) = 0$ is contained in $q + \Gamma_n$ for some $q \in \mathbb{R}^n$, where $c_k$ are real numbers not necessary to be non-negative and $\Gamma_n$ is the positive orthant, then this level set is convex. As an application, this result justifies the convexity of the level set of all general inverse $\sigma_k$ type equations, for example, the Monge--Amp\`ere equation, the Hessian equation, the J-equation, the deformed Hermitian--Yang--Mills equation, the special Lagrangian equation, etc. Moreover, we find a numerical condition to verify whether a level set of a general inverse $\sigma_k$ equation is contained in $q + \Gamma_n$ for some $q \in \mathbb{R}^n$, which is a way to determine the convexity of this level set.

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arXiv2022

The Deformed Hermitian--Yang--Mills Equation, the Positivstellensatz, and the Solvability

Let $(M, \omega)$ be a compact connected K\"ahler manifold of complex dimension four and let $[\chi] \in H^{1,1}(M; \mathbb{R})$. We confirmed the conjecture by Collins--Jacob--Yau [arXiv:1508.01934] of the solvability of the deformed Hermitian--Yang--Mills equation, which is given by the following nonlinear elliptic equation $\sum_{i} \arctan (\lambda_i) = \hat{\theta}$, where $\lambda_i$ are the eigenvalues of $\chi$ with respect to $\omega$ and $\hat{\theta}$ is a topological constant. This conjecture was stated in [arXiv:1508.01934], wherein they proved that the existence of a supercritical $C$-subsolution or the existence of a $C$-suboslution when $\hat{\theta} \in [ ( (n-2) + {2}/{n} ) {\pi}/{2}, n\pi/2 )$ will give the solvability of the deformed Hermitian--Yang--Mills equation. Collins--Jacob--Yau conjectured that their existence theorem can be improved when $\hat{\theta} \in ( (n-2 ) {\pi}/{2}, ( (n-2) + {2}/{n} ) {\pi}/{2} )$, where $n$ is the complex dimension of the manifold. In this paper, we confirmed their conjecture that when the complex dimension equals four and $\hat{\theta}$ is close to the supercritical phase $\pi$ from the right, then the existence of a $C$-subsolution implies the solvability of the deformed Hermitian--Yang--Mills equation.

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arXiv2020

Deformed Hermitian-Yang-Mills Equation on Compact Hermitian Manifolds

Let $(X, ω)$ be a compact connected Hermitian manifold of dimension $n$. We consider the Bott-Chern cohomology and let $[χ] \in H^{1,1}_{\text{BC}}(X; \mathbb{R})$. We study the deformed Hermitian-Yang-Mills equation, which is the following nonlinear elliptic equation $\sum_{i} \arctan (λ_i) = h(x)$, where $λ_i$ are the eigenvalues of $χ$ with respect to $ω$.

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