### Hessian measures. II.

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The convexity of level sets of solutions to the mean curvature equation is a long standing open problem. In this paper we give a counterexample to it.

We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the...

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