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Register a new userThe deformed Hermitian Yang-Mills equation on three-folds
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Added by
Vamsi Pritham Pingali
Publication date
2019
fields
Physics
and research's language is
English
Authors
Vamsi Pritham Pingali
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We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., $hat{theta} in (frac{pi}{2},frac{3pi}{2})$, on compact complex three-folds conditioned on a necessary subsolution condition. Our proof hinges on a delicate analysis of a new continuity path obtained by rewriting the equation as a generalised Monge-Amp`ere equation with mixed sign coefficients.
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In this paper, we consider the deformed Hermitian-Yang-Mills equation on closed almost Hermitian manifolds. In the case of hypercritical phase, we derive a priori estimates under the existence of an admissible $mathcal{C}$-subsolution. As an application, we prove the existence of solutions for the deformed Hermitian-Yang-Mills equation under the condition of existence of a supersolution.
We study a new deformed Hermitian Yang-Mills Flow in the supercritical case. Under the same assumption on the subsolution as Collins-Jacob-Yau cite{cjy2020cjm}, we show the longtime existence and the solution converges to a solution of the deformed Hermitian Yang-Mills equation which was solved by Collins-Jacob-Yau cite{cjy2020cjm} by the continuity method.
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This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills (HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with Kahler metrics $omega_epsilon$ when $epsilonto 0$. Here $omega_epsilon$ are flat and have areas $epsilon$ and $epsilon^{-1}$ on the two elliptic curves respectively. A family of Hermitian metrics on the vector bundle are explicitly constructed and with respect to them, the HYM metrics are normalized. We then compare the family of normalized HYM metrics with the family of constructed Hermitian metrics by doing estimates. We get the higher order estimates as long as the $C^0$-estimate is provided. We also get the estimate of the lower bound of the $C^0$-norm. If the desired estimate of the upper bound of the $C^0$-norm can be obtained, then it would be shown that these two families of metrics are close to arbitrary order in $epsilon$ in any $C^k$ norms.
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