We prove Liouville rigidity theorems for translation-invariant real and complex Hessian equations in the viscosity sense, where the PDE is encoded by an admissible set $\mathcal{A}$. The main structural notion is Liouville admissibility, a recursive geometric condition requiring each quotient set to be either boundary compatible or to fall into a terminal class. Our main theorem states that every bounded, globally $C^{0,\alpha}$ entire viscosity solution of \[ \mathrm{Hess}_{\mathbb F}u\in\partial\mathcal{A} \] is constant if and only if $\mathcal{A}$ is Liouville admissible; thus the Liouville-type property is characterized as a geometric property of the admissible set. A central class of examples arises from polarizations of univariate G{\aa}rding polynomials satisfying the monotone root sequence condition, producing mixed elementary-symmetric admissible sets and recovering the standard $k$-Hessian equations as monomial cases. The framework also allows anisotropic constructions, including linear pullbacks and intersections of admissible sets.
We prove a third-order derivative estimate for convex solutions to the real Monge-Amp\`ere equation ${\rm det}\,{\rm Hess}(u) = 1$ on an open set in $\mathbb{R}^{2m}$ under the additional assumption that ${\rm Hess}(u)$ lies in ${\rm Sp}(2m,\mathbb{R})$ at every point. Our method is a geometric interpretation and extension to higher dimensions of Nitsche's classical proof of the Bernstein theorem for the real Monge-Amp\`ere equation on $\mathbb{R}^2$. For $m = 1$ we also improve Nitsche's constant as well as some estimates due to Calabi, and we construct examples of solutions with interesting geometric behavior.
We study the quasilinear Liouville equation \[ -\Delta_n u=e^u \] on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Our first result shows that, if a solution $u$ satisfies the optimal logarithmic lower bound \[ u(x)\ge -\frac{n^2}{n-1}\log r(x)+o(\log r(x)) \quad \text{as }r(x)\to+\infty, \] then the underlying manifold is isometric to the Euclidean space and $u$ is a standard bubble solution. Both the leading coefficient and the remainder term in the assumption are sharp. The key ingredient in the proof is the connection between the logarithmic lower bound and a sharp upper bound on the total volume of the solution. We also formulate a conjecture concerning the interaction between the sub-logarithmic decay of solutions and the underlying geometry, and prove it for $n=2$, as well as for $n\ge 3$ under a strengthened assumption.
We prove a sharp partial regularity result for Hamiltonian stationary Lagrangian Lipschitz submanifolds in arbitrary smooth almost K\"ahler manifolds: every weak solution of the corresponding equation is smooth away from a relatively closed singular set of Hausdorff dimension at most $n-5$. We show that the estimate is optimal by constructing a nonzero two-homogeneous viscosity solution \[ U\in C^{1,1}(\mathbb{R}^5)\setminus C^2(\mathbb{R}^5) \] of the phase-zero special Lagrangian equation, whose level sets on $\mathbb{S}^4$ are the leaves of Cartan's isoparametric foliation. Its gradient graph is a non-flat calibrated cone, real analytic away from the vertex. This also gives the first $C^{1,1}$ but non-$C^2$ solution of the special Lagrangian equation, and shows that the same dimensional estimate is sharp in the case of special Lagrangian graphs.
Arunima Bhattacharya, Gerard Orriols, A. Skorobogatova· 0 citations
Let $(E,h)$ be a singular Hermitian vector bundle on a complex manifold $X$, and let \[ \mathcal E(h)_x=\{F\in\mathcal O(E)_x:|F|_h^2\in L^1_{\mathrm{loc},x}\} \] be its multiplier submodule sheaf. We introduce a finite plurisubharmonic Gram scalarization condition under which the higher-rank integrability problem reduces to finitely many scalar multiplier ideals. This reduction yields coherence and strong openness without imposing a general positivity hypothesis. When the scalar weights and the varying weight have analytic singularities, it also gives a theory of module jumping numbers, including a simultaneous-residue criterion for actual jumps. Finally, we study the induced Skoda filtration: Artin--Rees yields eventual periodicity, Tor controls whether periodicity starts at the scalar threshold, and a direct-image quotient measures the obstruction to descent.
We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0 \qquad\text{in }\mathbb R^n. \] If $n=2$ and $p\geqslant1$, or if $n\geqslant3$ and $1\leqslant p<\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ vanishes identically. No symmetry, decay, integrability, or uniform spacelike gap is assumed. A key independent ingredient is a universal bound, valid for every $n\geqslant2$ and $p\geqslant1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u|^2)^{-1/2}$. Thus pointwise strict spacelikeness automatically improves to a uniform spacelike gap, including in the critical and supercritical regimes. The proof combines a geometric Bernstein estimate, comparison with an explicit hyperbolic cap, and weighted trace-free tensor identities. The result extends the known radial nonexistence theorem to arbitrary entire solutions and yields a geometric half-space rigidity theorem for complete spacelike hypersurfaces.
Xi-nan Ma, Tian Wu, Wangzhe Wu et al.· 0 citations
We study regularity, Harnack inequalities, Liouville rigidity, and principal eigenvalues for viscosity solutions of singular or degenerate fully nonlinear elliptic equations $\Phi(x,|\nabla u|)F(D^2u)-H(x,\nabla u)+c(x)|u|^{i(\Phi)}u=h(x)$ in a bounded domain $\Omega$, where $\Phi$ describes the singular or degenerate dependence on the gradient and $H$ is a Hamiltonian. We first prove global $C^{1,\gamma}$ regularity for the Dirichlet problem without lower-order terms. The argument combines a global $L^\infty$ estimate from the Alexandroff--Bakelman--Pucci inequality, boundary barriers yielding a global Lipschitz bound, and a compactness-based iterative approximation scheme. We next establish an additive Harnack inequality for nonnegative viscosity solutions by sliding from below a cusp function of the form $-|x|^{1/2}$. Under an additional homogeneity assumption, this yields the classical Harnack inequality and Liouville-type theorems in $\mathbb{R}^n$. Finally, under suitable homogeneity and comparison assumptions, we develop a generalized Dirichlet principal-eigenvalue theory for the full operator. We prove the existence of principal eigenfunctions and characterize the associated eigenvalues through maximum and minimum principles. These results provide a unified framework for global regularity, Harnack estimates, Liouville rigidity, and principal eigenvalues for a broad class of singular and degenerate fully nonlinear equations with Hamiltonian terms.
Trung-Hieu Huynh, T. Khuu, Hoang-Hung Vo· 0 citations