Preprint
Determinantal capacity andL∞ Estimates
arXiv
arXiv
09/21/2026
DOI: 10.48550/arxiv.2609.24078
Abstract
We introduce determinantal ellipticity, or det-ellipticity, a quantitative structural condition for fully nonlinear elliptic operators on compact Kähler manifolds that provides the link between the L∞-estimates and algebraic/combinatorial properties of a large class of Hessian elliptic operators.
On the analytic side, we introduce D-subsolutions extending the determinant sublevel condition of Sui–Sun. Using the auxiliary comparison method of Guo-Phong–Tong and pluripotential theory for complex Monge–Ampère equations, we obtain relative L∞-estimates in big cohomology classes under a determinant-entropy bound for viscosity supersolutions and singular reference potentials. Det-ellipticity provides a systematic construction of such subsolutions.
On the algebraic side, we study Gårding elliptic polynomial operators of degree d. We characterize the determinant increment bound with exponent d/n by positivity of the determinantal capacity of the top homogeneous part. This is equivalent to balanced-point conditions for the Newton polytopes of rank-one scalarizations and to slope semistability of an associated subspace polymatroid. Together with the analytic hypotheses, these criteria yield relative L∞ estimates for the corresponding fully nonlinear equation
Details
- Title: Subtitle
- Determinantal capacity andL∞ Estimates
- Creators
- Hao FangBiao MaJinyang Wu
- Resource Type
- Preprint
- Publication Details
- arXiv
- DOI
- 10.48550/arxiv.2609.24078
- ISSN
- 2331-8422
- Publisher
- arXiv
- Language
- English
- Date posted
- 09/21/2026
- Academic Unit
- Mathematics
- Record Identifier
- 9985236417502771
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