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Let $(M,g)$ be an oriented compact Riemannian $4$-manifold with boundary $\partial M$. Let $a\in \Omega^1$ such that $*a\big|_{\partial M}=0$, i.e. $a(\nu)=0$ on $\partial M$, where $\nu$ denotes the unit normal field.

Question. If $d^+a=0$, can we get $da=0$?

PS:

  • If $M$ is closed, we know it is true, by $d(a\wedge da)=|d^+a|^2-|d^-a|^2$ and Stokes formula.

  • If $a\big|_{\partial M}=0$, by the same method, we know it is still true.

The reason to ask such a question: Under the same condition, we have that $$\|a\|_{L^{p}_1}\leq C(\|da\|_{L^p}+\|d^*a\|_{L^p}+\|a\|_{L^p}),$$ by Theorem 5.1 of Katrin Wehrheim, Uhlenbeck Compactness, Princeton University, Princeton, NJ. I wonder can we have a similar estimate.

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No. For example, take $M$ to be the Euclidean ball with $$ a=-x^1 dx^0+x^0 dx^1+x^3 dx^2-x^2 dx^3. $$

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