In this talk I will report two recent results on the conformal k-Hessian equation on manifolds of dimension n>2. The first one, with W.M. Sheng and N.S. Trudinger, is the existence of solutions to the so-called k-Yamabe problem for k less than or equal to n/2, assuming that the metric is admissible and the equation is variational. The second one, with N.S. Trudinger, is the compactness of admissible metrics to the conformal k-Hessian equation when k>n/2, which also yields the existence of solutions to the k-Yamabe problem if there is an admissible metric, first proved by Gursky and Viaclovsky.

Lecture #12236

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