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Rirong Yuan

Publications and source records attributed to Rirong Yuan.

At least 19 recordsLinked to original sources
arXiv2026

On a continuity method for Dirichlet problem of Hessian equations

In this paper, we develop a continuity method for the Dirichlet problem of Hessian equations on Riemannian manifolds. Such equations, introduced by Caffarelli, Nirenberg and Spruck, are defined in terms of the eigenvalues of the Hessian and a given pair $(f,Γ)$, where $f$ is a symmetric function defined in a symmetric cone $Γ\subset\mathbb{R}^n$, and $Γ$ specifies the set of admissible eigenvalues for the solution. Our method combines techniques from Morse theory with a characterization of the pair $(f,Γ)$. More precisely, in the type 2 case, we first construct admissible functions using Morse theory, and then solve the Dirichlet problem without any additional assumptions on the boundary or the subsolution. Building on this characterization of the pair, we can approximate the type 1 equation by a family of type 2 equations.

math.AP↗
arXiv2025

The structure of fully nonlinear equations and its applications to prescribed problems on complete conformal metrics

This paper investigates the structure of fully nonlinear equations and their applications to geometric problems. We solve some fully nonlinear version of the Loewner-Nirenberg and Yamabe problems. Notably, we introduce Morse theory techniques to construct admissible metrics under a weak condition on the underlying metric, which can be further relaxed in a broad setting. Furthermore, we provide some topological obstruction to demonstrate the optimality of our structural conditions.

math.AP↗
arXiv2025

Fully non-linear elliptic equations on complex manifolds

In this paper, we study a broad class of fully nonlinear elliptic equations on Hermitian manifolds. On one hand, under the optimal structural assumptions we derive $C^{2,α}$-estimate for solutions of the equations on closed Hermitian manifolds. On the other hand, we treat the Dirichlet problem. In both cases, we prove the existence theorems with unbounded condition.

math.AP↗
arXiv2023

Notes on conformal metrics of negative curvature on manifolds with boundary

We use certain Morse functions to construct conformal metrics such that the eigenvalue vector of modified Schouten tensor belongs to a given cone. As a result, we prove that any Riemannian metric on compact 3-manifolds with boundary is conformal to a compact metric of negative sectional curvature.

math.DG↗
arXiv2023

An extension of prescribed problems on the conformal classes of complete metrics

We solve prescribed problems for modified Schouten tensors in the conformal classes of smooth complete metrics, which extends the results obtained in prequel \cite{yuan-PUE1}. The key ingredient is to confirm the uniform ellipticity of operators under an assumption, which is sharp as shown by obstructions from topology and function theory.

math.DG↗
arXiv2022

On the conformal bending of a closed Riemannian manifold

In this paper, we bend a closed Riemannian manifold in the conformal class, through solving a fully nonlinear equation. As a result, we prove that each metric of quasi-negative Ricci curvature is conformal to a metric with negative Ricci curvature.

math.DG↗
arXiv2022

The partial uniform ellipticity and prescribed problems on the conformal classes of complete metrics

We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully nonlinear equations of elliptic and parabolic type. As applications, we solve a fully nonlinear version of the Loewner-Nirenberg problem and a noncompact complete version of fully nonlinear Yamabe problem. Our method is delicate as shown by a topological obstruction.

math.DG↗
arXiv2022

On the regularity of Dirichlet problem for fully non-linear elliptic equations on Hermitian manifolds

We derive the solvability and regularity of the Dirichlet problem for fully non-linear elliptic equations possibly with degenerate right-hand side on Hermitian manifolds, through establishing a quantitative version of boundary estimate under a subsolution assumption. In addition, we construct the subsolution when the background manifold is a product of a closed Hermitian manifold with a compact Riemann surface with boundary.

math.AP↗
arXiv2022

Regularity of fully non-linear elliptic equations on Hermitian manifolds

In this paper we propose new insights and ideas to set up quantitative boundary estimates for solutions to Dirichlet problem of a class of fully non-linear elliptic equations on compact Hermitian manifolds with real analytic Levi flat boundary. With the quantitative boundary estimates at hand, we can establish the gradient estimate and give a unified approach to investigate the existence and regularity of solutions of Dirichlet problem with sufficiently smooth boundary data, which include the geodesic equation in the space of Kähler metrics as a special case. Our method can also be applied to Dirichlet problem for analogous fully non-linear elliptic equations on a compact Riemannian manifold with concave boundary.

math.AP↗
arXiv2021

On the partial uniform ellipticity and complete conformal metrics with prescribed curvature functions on manifolds with boundary

We consider the problem of finding complete conformal metrics with prescribed curvature functions of the Einstein tensor and of more general modified Schouten tensors. To achieve this, we reveal an algebraic structure of a wide class of fully nonlinear equations. Our method is appropriate and delicate as shown by a topological obstruction. Finally, we discuss Hessian equations and Weingarten equations by confirming a key assumption.

math.DG↗
arXiv2021

Local $C^0$-estimate and existence theorems for some prescribed curvature problems on complete noncompact Riemannian manifolds

In this article we study a class of prescribed curvature problems on complete noncompact Riemannian manifolds. To be precise, we derive local $C^0$-estimate under an asymptotic condition which is in effect optimal, and prove the existence of complete conformal metrics with prescribed curvature functions. A key ingredient of our strategy is Aviles-McOwen's result or its fully nonlinear version on the existence of complete conformal metrics with prescribed curvature functions on manifolds with boundary.

math.DG↗
arXiv2020

Regularity of fully non-linear elliptic equations on Hermitian manifolds. II

In this paper we investigate the regularity and solvability of solutions to Dirichlet problem for fully non-linear elliptic equations with gradient terms on Hermitian manifolds, which include among others the Monge-Ampère equation for $(n-1)$-plurisubharmonic functions. Some significantly new features of regularity assumptions on the boundary and boundary data are obtained, which reveal how the shape of the boundary influences such regularity assumptions. Such new features follow from quantitative boundary estimates which specifically enable us to apply a blow-up argument to derive the gradient estimate. Interestingly, the subsolutions are constructed when the background space is moreover a product of a closed Hermitian manifold with a compact Riemann surface with boundary.

math.AP↗