We develop a pluripotential approach to complex Hessian equations on compact Hermitian manifolds. In this setting, the lack of closedness of the background metric introduces torsion terms that prevent a direct extension of the K\"ahler theory. Our main result is a uniform $L^\infty$ estimate for bounded $\omega$-$m$-subharmonic solutions of the equation \[ (\omega + dd^c u)^m \wedge \omega^{n-m} = cf\,\omega^n, \] under the assumption that $f \in L^p$, $f \ge 0$ for some $p>1$. The proof combines a weak comparison principle with torsion error, a capacity theory adapted to the Hermitian setting, and a nonlinear iteration scheme controlling the decay of sublevel sets. As applications, we obtain existence, stability and compactness results for weak solutions with $L^p$ densities. These results extend several aspects of the pluripotential theory. of complex Hessian equations beyond the K\"ahler framework.
We study complex $m$-Hessian equations on bounded hyperconvex domains with right-hand side in $L^1(\Omega)$. The main contribution of this paper is a strong stability result for weak solutions in the Hessian energy sense. More precisely, if $f_j \to f$ in $L^1(\Omega)$ and $u_j, u$ are the corresponding solutions, then \[ \int_\Omega |u_j-u|\, H_m(u_j) \longrightarrow 0. \] This provides convergence in the natural energy topology associated to the Hessian operator, which is significantly stronger than convergence in capacity. For completeness, we also recall the existence of solutions and stability in capacity, which follow from known results in the literature.
Let $(L,e^{-\phi})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and put $\omega=\ddbar\phi$. We obtain effective pointwise estimates for the Bergman form of $H^0(X,K_X\otimes L^m)$. If $\Ric\omega\leq\omega$ and the shortest nonconstant closed geodesic has length at least $2\pi$, then \[ K_{m\phi}\geq \frac{2m-1}{4\pi}\,\omega, \] and the constant is sharp on $(\mathbb P^1,\mathcal O_{\mathbb P^1}(2))$. A local version, depending on an upper curvature bound and the injectivity radius, recovers the first two terms of the Bergman expansion when the curvature is constant. Under the two-sided bound $-\omega\leq\Ric\omega\leq\omega$ and the same closed-geodesic hypothesis, we also prove \[ K_{m\phi}\leq \frac{m\omega}{2\pi} \left(1+\frac{54.8\log(2m)}{m-\frac{1}2}\right). \] The lower estimates use the deformation-to-the-tangent-space form of the Ohsawa--Takegoshi theorem established by He, Wang, and the author, whereas the upper bound combines a weighted submean inequality with quantitative isothermal coordinates which was obtained in recent work by Eilat.
In this paper, we establish interior $C^2$ estimates for admissible semiconvex solutions to the general Hessian quotient equation $\frac{\sigma_k}{\sigma_l}(D^2u)=f(x,u),$ for the cases $l=k-1$ and $l=k-2$, where $f$ is a positive $C^2$ function. Such estimates are known to fail in general for $k-l\geq 3$, even for convex solutions, as shown by counterexamples due to Lu \cite{LuGeneral}. The main ingredient is a quantitative concavity inequality for the Hessian quotient operator under the semiconvex condition. Our result provides a unified argument to such general Hessian quotient equations for $2\leq k\leq n-1$ in arbitrary dimensions.
Let $2\le k\le n$, let $\Omega\subset\mathbb{R}^n$ be open and convex, and let $u$ be a convex viscosity solution of $\sigma_k(D^2u)=1$ in $\Omega$. We prove that the set on which $u$ fails to be locally $C^2$ has vanishing $(n-1)$-dimensional Hausdorff measure. In the intermediate range $3\le k<n$, this gives a codimension-one refinement of the known almost-everywhere partial regularity, and the exponent is sharp. More generally, for a convex viscosity subsolution of $\sigma_k(D^2u)\ge\lambda>0$, we obtain Hausdorff bounds for strata defined by the affine dimension of all supporting contact sets. The proof combines a support-dependent Chou--Wang barrier argument, an estimate for the product of the smallest $k$ semiaxes of a John ellipsoid, and Mooney's convex section-covering theorem. As a direct analytical consequence, the full distributional Hessian is absolutely continuous and $u\in W^{2,1}_{\mathrm{loc}}(\Omega)$, yielding a $k$-Hessian counterpart of the $W^{2,1}$ regularity known for singular Monge--Amp\`ere solutions. In a logically separate structural part, we characterize the distinguished number of flat directions, $n-k+1$, by an asymptotic infimum mean-value formula over affine sections, and explain how this mean-value heuristic leads to the supporting-contact geometry used in the proof.
Let $n\ge2$ and let $u$ be a smooth 2-convex and semi-convex solution of \[ \frac{\sigma _2(D^2u)}{\sigma _1(D^2u)}=f(x). \] We prove an interior Hessian estimate depending on the Lipschitz norm of $f$. The proof combines the integral approach of Chen--Jian--Zhou with the algebraic reduction of the quotient equation to a $\sigma _2$ structure. The main new point is a shifted algebraic inequality that yields a shifted trace Jacobi inequality in divergence form for $\log(\Delta u+a)$. We work with the linearized operator $G=(\Delta u-f)I-D^2u$ of the equivalent equation $\sigma_2(D^2u)=f \Delta u$. The almost divergence-free identity \(\partial_iG_{ij}=-f_j\) enables us to control the \(\Delta f\) term by integration by parts solely in terms of the Lipschitz norm of \(f\). A Legendre--Lewy transformation converts the resulting degenerate divergence-form equation into a uniformly elliptic one. The estimate then follows from a mean-value inequality together with a weighted energy argument. As an application, in dimension two we obtain interior $C^2$ regularity for convex viscosity solutions with positive Lipschitz right-hand side. Moreover, our counterexamples show that the Lipschitz regularity required of the right-hand side is optimal.
Let $(M,\omega)$ be a compact Hermitian manifold, and let $\Gamma$ be a symmetric convex cone. We develop a quantitative regularization method for $\Gamma$-admissible functions. As an application, we prove H\"older continuity for every pluripotential solution of complex $m$-Hessian equations whose right-hand sides belong to $L^p$, for $p>\frac{n}{m}$. These results extend to a more general class of complex Hessian equations satisfying the structural condition used by Guo--Phong--Tong.