The $L_p$ dual Christoffel-Minkowski type problem for a class of Hessian quotient equations

Abstract: In this paper, we investigate an $L_p$ dual Christoffel-Minkowski type problem for the Hessian quotient operator $\frac{σ{k}(Λ)}{σ{l}(Λ)}$, where the operator $Λ$ has been widely studied in the literature. Exploiting the recently discovered ``inverse convexity'' property of this class of operators, we establish a full rank theorem under suitable structural assumptions. Together with a priori estimates, this result enables us to prove the existence and uniqueness of strictly spherically convex solutions to the above $L_p$ dual Christoffel-Minkowski type problem.

• The paper demonstrates that the inverse convexity property of the Hessian quotient operator enables full C^2 estimates, overcoming critical third-order challenges.
• It establishes existence and uniqueness of strictly spherically convex solutions via the method of continuity and precise a priori estimates in both homogeneous and nonhomogeneous cases.
• The analytical framework leveraging admissible cones and eigenvalue techniques generalizes classical Minkowski problems to a broader class of geometric PDEs.

The $L_p$ Dual Christoffel-Minkowski Type Problem for Hessian Quotient Equations

Introduction and Problem Overview

This paper addresses a highly technical extension of the dual Brunn-Minkowski theory, focusing on the $L_p$ dual Christoffel-Minkowski problem for a broad class of Hessian quotient equations. The central object of study is the PDE

$$\frac{\sigma_k(\Lambda(\nabla^2 u + uI))}{\sigma_l(\Lambda(\nabla^2 u + uI))} = u^{p-1}(u^2 + |\nabla u|^2)^{\frac{k+1-q}{2}} \varphi(x) \quad \text{on}~\mathbb{S}^n,$$

where

• $\sigma_k$ denotes the $k$-th elementary symmetric polynomial,
• $\Lambda(\cdot)$ is the vector of $\mathscr{P}$-eigenvalues (sums over ordered multi-index sets) of the perturbed Hessian,
• $p,q \in \mathbb{R}$ and $l<k\leq N$ (with $N$ combinatorially determined by $L_p$0),
• $L_p$1 is smooth and strictly positive.

This formulation generalizes multiple classical Minkowski and Christoffel-Minkowski problems, subsuming them via specific choices of parameters and the eigenvalue operator. The case $L_p$2 leads to new geometric situations not encountered in classical convex body PDEs.

Technical Contributions

Inverse Convexity and Full Rank Theorem

A crucial and novel element in the analysis is the identification and exploitation of an "inverse convexity" property of the Hessian quotient operator in question. Specifically, for $L_p$3 in an appropriate range, the operator $L_p$4 is shown to be inverse convex with respect to the underlying matrix argument. This enables highly nontrivial handling of third-order terms in $L_p$5 estimates, which is generally the key obstacle for Hessian quotient type fully nonlinear PDEs. The full rank theorem (Theorem 3.1) asserts that admissible solutions with semi-definite spherically convex matrices are indeed strictly spherically convex under explicit analytic assumptions on $L_p$6 and the parameters.

Existence and Uniqueness (Nonhomogeneous and Homogeneous Cases)

Via the method of continuity and a priori estimates, existence and uniqueness of strictly spherically convex solutions are obtained in both nonhomogeneous ($L_p$7) and homogeneous ($L_p$8) regimes (Theorems 1.2, 1.3, 1.4):

• The authors provide sup/inf ($L_p$9), gradient ($$\frac{\sigma_k(\Lambda(\nabla^2 u + uI))}{\sigma_l(\Lambda(\nabla^2 u + uI))} = u^{p-1}(u^2 + |\nabla u|^2)^{\frac{k+1-q}{2}} \varphi(x) \quad \text{on}~\mathbb{S}^n,$$0), and Hessian ($$\frac{\sigma_k(\Lambda(\nabla^2 u + uI))}{\sigma_l(\Lambda(\nabla^2 u + uI))} = u^{p-1}(u^2 + |\nabla u|^2)^{\frac{k+1-q}{2}} \varphi(x) \quad \text{on}~\mathbb{S}^n,$$1) bounds, with constants explicitly depending on the problem data.
• The strong monotonicity with respect to the unknown and uniqueness are handled by a careful maximum principle argument exploiting the structure of the right hand side and the monotonicity of the quotient of symmetric functions.
• For the homogeneous case ($$\frac{\sigma_k(\Lambda(\nabla^2 u + uI))}{\sigma_l(\Lambda(\nabla^2 u + uI))} = u^{p-1}(u^2 + |\nabla u|^2)^{\frac{k+1-q}{2}} \varphi(x) \quad \text{on}~\mathbb{S}^n,$$2), the authors introduce an auxiliary scaling parameter and approximate via nonhomogeneous equations, passing to the limit using uniform estimates that are independent of the approximation parameter.

Analytical Framework and Admissible Cones

The problem is set up in the context of eigenvalues living inside $$\frac{\sigma_k(\Lambda(\nabla^2 u + uI))}{\sigma_l(\Lambda(\nabla^2 u + uI))} = u^{p-1}(u^2 + |\nabla u|^2)^{\frac{k+1-q}{2}} \varphi(x) \quad \text{on}~\mathbb{S}^n,$$3-admissible cones, corresponding to the positivity of symmetric polynomials of the $$\frac{\sigma_k(\Lambda(\nabla^2 u + uI))}{\sigma_l(\Lambda(\nabla^2 u + uI))} = u^{p-1}(u^2 + |\nabla u|^2)^{\frac{k+1-q}{2}} \varphi(x) \quad \text{on}~\mathbb{S}^n,$$4-eigenvalues. The interplay between these cones, the linear action of the derivation operator on exterior powers, and their monotonicity/concavity structure is systematically utilized. This framework enables the paper to capture convexity preservation along the deformation path and the nondegeneracy of the linearized operator.

Strong Regularity and Geometric Implications

The regularity obtained is classical: solutions are shown to be smooth on $$\frac{\sigma_k(\Lambda(\nabla^2 u + uI))}{\sigma_l(\Lambda(\nabla^2 u + uI))} = u^{p-1}(u^2 + |\nabla u|^2)^{\frac{k+1-q}{2}} \varphi(x) \quad \text{on}~\mathbb{S}^n,$$5 under smooth data. The strict spherical convexity conclusion guarantees that the support function arises from a genuinely strictly convex body. The results extend uniqueness and regularity theorems for the Christoffel-Minkowski and classical Minkowski-type problems to the setting of quotient equations associated with linear derivations of symmetric matrices.

Main Theorems and Claims

• For $$\frac{\sigma_k(\Lambda(\nabla^2 u + uI))}{\sigma_l(\Lambda(\nabla^2 u + uI))} = u^{p-1}(u^2 + |\nabla u|^2)^{\frac{k+1-q}{2}} \varphi(x) \quad \text{on}~\mathbb{S}^n,$$6 and $$\frac{\sigma_k(\Lambda(\nabla^2 u + uI))}{\sigma_l(\Lambda(\nabla^2 u + uI))} = u^{p-1}(u^2 + |\nabla u|^2)^{\frac{k+1-q}{2}} \varphi(x) \quad \text{on}~\mathbb{S}^n,$$7 satisfying a precise analytic pinching condition, there exists a unique, strictly spherically convex, positive solution (Theorem 1.2).
• In the homogeneous degenerate case $$\frac{\sigma_k(\Lambda(\nabla^2 u + uI))}{\sigma_l(\Lambda(\nabla^2 u + uI))} = u^{p-1}(u^2 + |\nabla u|^2)^{\frac{k+1-q}{2}} \varphi(x) \quad \text{on}~\mathbb{S}^n,$$8, for any positive smooth $$\frac{\sigma_k(\Lambda(\nabla^2 u + uI))}{\sigma_l(\Lambda(\nabla^2 u + uI))} = u^{p-1}(u^2 + |\nabla u|^2)^{\frac{k+1-q}{2}} \varphi(x) \quad \text{on}~\mathbb{S}^n,$$9, there exists a unique (up to scaling/dilation) strictly spherically convex positive solution (Theorem 1.3).
• For the case $\sigma_k$0, both existence and uniqueness are established, and explicit sharp estimates are derived (Theorem 1.4).
• A new and essential property of the Hessian quotient operator is established: the strong inverse convexity and monotonicity with respect to matrix arguments, allowing full $\sigma_k$1 estimates without convexity loss along the method of continuity.

Methodological Distinction and Analytical Rigor

The proofs are grounded in a blend of geometric and analytic tools, including:

• Microscopic convexity principles adapted from Bian-Guan,
• The Evans-Krylov theorem for higher regularity,
• A scaling-invariant approach in the homogeneous regime,
• Fully nonlinear PDE theory on admissible cones, extending previous works by Caffarelli-Nirenberg-Spruck, Guan, and colleagues to the more intricate Hessian quotient context.

The paper introduces multi-index notation for eigenvalue analysis, adapts classical majorization and monotonicity results to the context of these generalized Hessian operators, and leverages their concavity/convexity properties in the crucial estimates.

Theoretical and Practical Implications

The results represent a significant generalization of $\sigma_k$2-type Minkowski problems, proving existence, uniqueness, and strict convexity for a categorically larger class of geometric PDEs. The analytic techniques for handling quotient operators associated with exterior algebra actions may prove instrumental in other fully nonlinear geometric PDEs, notably in calibrated geometry and in problems related to Gauduchon-type metrics in complex analysis.

Practical implications lie in the characterization of convex bodies via prescribed measures not only of curvature but of more elaborate symmetric quotients—enabling new flows, functional inequalities, and geometric optimization problems in convex geometry and PDE.

Potential Future Developments

• The framework suggests possible extensions to more general geometric settings (Riemannian or Hermitian manifolds, other symmetric spaces).
• Analytic and geometric inequalities arising from the sharp estimates in this setting could provide new insights for Brunn-Minkowski theory and its duals.
• The inverse convexity property may be useful for geometric flows and evolution equations for general curvature quotients.
• Potential extension to degenerate (boundary or singular) data, or to non-smooth geometric configurations, exploiting the structure of admissible cones and monotonicity phenomena.

Conclusion

This work rigorously establishes the existence, uniqueness, and regularity of strictly spherically convex solutions to the $\sigma_k$3 dual Christoffel-Minkowski problem for a class of Hessian quotient equations. The main analytical innovation—a strong inverse convexity principle for the quotient operator—enables the adaptation of the method of continuity to the highly nontrivial, nonstandard setting of $\sigma_k$4-eigenvalues. The solutions provided unify and generalize a large collection of central problems in convex and differential geometry, and the techniques developed are expected to influence further research in geometric analysis and fully nonlinear PDEs.

Reference: "The $\sigma_k$5 dual Christoffel-Minkowski type problem for a class of Hessian quotient equations" (2604.10924).