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Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.

2 votes
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113 views

Is this $1$-form harmonic?

Let $(M^3,g)$ be a compact, connected and oriented Riemannian $3$-manifold with boundary. For a harmonic map $u : M \to \mathbb{S}^1$ satisfying Neumann condition along $\partial M$, let $h = u^*(d \t …