## Journal of Differential Geometry

- J. Differential Geom.
- Volume 88, Number 3 (2011), 395-482.

### Dirichlet Duality and the Nonlinear Dirichlet Problem on Riemanninan Manifolds

F. Reese Harvey and H. Blaine Lawson, Jr.

#### Abstract

In this paper we study the Dirichlet problem for fully nonlinear
second-order equations on a riemannian manifold. As in our
previous paper, "Dirichlet duality and the non-linear
Dirichlet problem," *Comm. on Pure and Applied Math.*, 62 (2009), 396–443, we define equations via closed subsets of
the 2-jet bundle where each equation has a natural dual equation.
Basic existence and uniqueness theorems are established in a wide
variety of settings. However, the emphasis is on starting with a
constant coefficient equation as a model, which then universally
determines an equation on every riemannian manifold which is
equipped with a topological reduction of the structure group to
the invariance group of the model. For example, this covers all
*branches* of the homogeneous complex Monge-Ampère equation
on an *almost complex* hermitian manifold $X$.

In general, for an equation $F$ on a manifold $X$ and a smooth domain $\Omega \subset\subset X$, we give geometric conditions which imply that the Dirichlet problem on $\Omega$ is uniquely solvable for all continuous boundary functions. We begin by introducing a weakened form of comparison which has the advantage that local implies global. We then introduce two fundamental concepts. The first is the notion of a monotonicity cone $M$ for $F$. If $X$ carries a global $M$-subharmonic function, then weak comparison implies full comparison. The second notion is that of boundary $F$-convexity, which is defined in terms of the asymptotics of $F$ and is used to define barriers. In combining these notions the Dirichlet problem becomes uniquely solvable as claimed.

This article also introduces the notion of *local affine jet-equivalence*
for subequations. It is used in treating the cases above,
but gives results for a much broader spectrum of equations on
manifolds, including inhomogeneous equations and the Calabi-Yau
equation on almost complex hermitian manifolds.

A considerable portion of the paper is concerned with specific examples. They include a wide variety of equations which make sense on any riemannian manifold, and many which hold universally on almost complex or quaternionic hermitian manifolds, or topologically calibrated manifolds.

#### Article information

**Source**

J. Differential Geom., Volume 88, Number 3 (2011), 395-482.

**Dates**

First available in Project Euclid: 15 November 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1321366356

**Digital Object Identifier**

doi:10.4310/jdg/1321366356

**Mathematical Reviews number (MathSciNet)**

MR2844439

**Zentralblatt MATH identifier**

1235.53042

#### Citation

Harvey, F. Reese; Lawson, H. Blaine. Dirichlet Duality and the Nonlinear Dirichlet Problem on Riemanninan Manifolds. J. Differential Geom. 88 (2011), no. 3, 395--482. doi:10.4310/jdg/1321366356. https://projecteuclid.org/euclid.jdg/1321366356