So I am unsure of how the hessian becing less than or equal zero contradicts that the laplacian is greater than zero. Can someone provide some input for the reasoning. Thanks.
2026-04-29 18:17:32.1777486652
Strong maximum principle on Riemannian manifolds
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The Laplacian is the trace of the Hessian, which is positive. However the Hessian is symmetric, and symmetric plus negative semidefinite implies nonpositive trace (this can be seen easily through diagonalization).