# Minimizing area in relative homology class

A well known result in geometric measure theory asserts that if $$(M^{n+1}, g)$$ is a closed Riemannian manifold and $$\alpha \in H_n(M)$$ is a nonzero homology class, then there exists a closed embedded minimal hypersurface $$\Sigma$$ (smooth outside a singular set of dimension less than or equal to $$n-7$$) minimizing area in $$[\Sigma]=\alpha$$.

My question is if this result can be (or has been) generalized for manifolds with nonempty boundary. Precisely, I wonder if the following is true:

Let $$(M^{n+1}, g)$$ be a compact Riemannian manifold with nonempty boundary. Given $$\alpha \in H_n(M, \partial M)$$ a nonzero relative homology class, there exists an embedded and free boundary minimal hypersurface $$\Sigma$$ (smooth outside a singular set of dimension less than or equal to $$n-7$$) minimizing area in $$[\Sigma]=\alpha$$.

• Yes, it works: double your manifold across the boundary and observe that the classical argument works equivariantly, with respect to the natural involution of the double. Feb 17, 2020 at 14:02
• Will the result surface be free boundary? Feb 17, 2020 at 16:12
• Doubling a manifold with boundary results in a manifold without boundary. Thus, the double of $\Sigma$ has no boundary. Feb 17, 2020 at 16:54
• Yes: Compare it with a competitor $\Sigma_1\subset M$. Then $Area(\Sigma_1)=Area(D\Sigma_1)/2\ge Area(D\Sigma_0)/2=Area(\Sigma_0)$, where $D\sigma_i$ is the double of $\Sigma_i$ inside $DM$, the double of $M$. Feb 17, 2020 at 19:17
• Well, since it is an area-minimizer in its homology class.... Feb 18, 2020 at 2:29