Corollary 8.4.7`BEA27E46DAEE`

Corollary 8.4.7.BEA27E46DAEE Under the hypotheses of the proposition, one has

\[ \sup _ X|\partial u|_ g+\sup _ X|\partial \bar\partial u|_ g\leq C, \]

with a constant depending on the same fixed data and no derivatives of $v$.

Proof. The Hessian density is $F(W)^ m\leq f_1^ m$. Apply the trace-to-gradient blowup proposition to the fixed background $\theta =\omega $ and the preceding trace estimate. It gives the uniform gradient bound. The trace is then uniformly bounded, and the $\Gamma _2$ identity bounds the absolute value of every eigenvalue of $W$ by its trace. Subtracting the identity matrix gives the complex-Hessian bound. $\square$

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BibTeX citation

@misc{pome-tag-BEA27E46DAEE,
  author = {{Pome contributors}},
  title = {Corollary 8.4.7},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/BEA27E46DAEE/}
}

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Corollary 8.4.7BEA27E46DAEE

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Corollary 8.4.7`BEA27E46DAEE`