On compact Riemannian manifolds the Bourguignon Laplacian has finite-dimensional kernel of Codazzi tensors with constant trace; under nonnegative sectional curvature such tensors are parallel, and the paper gives eigenvalue lower bounds.
and Shoen R., Lp and mean value properties of subharmonic func- tions on Riemannian manifolds, Acta Mathematica, 153:1 (1984), 279 – 301
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The Bourguignon Laplacian and harmonic symmetric bilinear forms
On compact Riemannian manifolds the Bourguignon Laplacian has finite-dimensional kernel of Codazzi tensors with constant trace; under nonnegative sectional curvature such tensors are parallel, and the paper gives eigenvalue lower bounds.