On a class of Hessian type equations on Riemannian manifolds
Heming Jiao, Jinxuan Liu
Abstract
In this paper, we consider a class of Hessian type equations which include the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(n-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Monge-Ampère equation on Riemannian manifolds. The <italic>a priori</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C squared"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> estimates and the existence of solutions are established.