# Manifolds with boundary admitting no closed embedded minimal hypersurface

The following Theorem is proved in the paper entitled "Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary", by A. Fraser and M. Li:

Let $$M^n$$ be a compact $$n$$-dimensional Riemannian manifold with nonempty boundary $$\partial M$$. Suppose $$M$$ has nonnegative Ricci curvature and the boundary $$\partial M$$ is strictly mean convex with respect to the inward unit normal. Then, $$M$$ contains no smooth, closed, embedded minimal hypersurface.

My question is: what are examples of compact Riemannian $$3$$-manifolds with nonnegative scalar curvature (but not nonnegative Ricci curvature) and mean convex boundary that don't admit closed embedded minimal surfaces?

A solid torus should work. Choose cylindrical coordinates $$(r,\theta, \lambda)$$, $$0\leq r \leq r_0 < \pi/2, 0\leq \theta \leq 2\pi, 0\leq \lambda \leq l$$, where we equate $$(r,\theta,0)\sim (r,\theta, l)$$ and $$(0,\theta, \lambda)\sim (0,0,\lambda)$$. Put a Riemannian metric on this solid torus of the form $$dr^2+ f(r)^2 d\theta^2 + g(r)^2 d\lambda^2$$, and $$f(r)=\sin(r), g(r)=\cosh(\epsilon r)$$, where $$0 < \epsilon$$ is small.
The sectional curvatures of such a metric are computed in Lemma 2.3 of this paper as $$K_{\theta\lambda}=-\frac{f'g'}{fg},\ K_{r\theta}=-\frac{f''}{f},\ K_{r\lambda}=-\frac{g''}{g}$$
and the mean curvature of the level $$r$$ torus is $$\frac12(\frac{f'}{f}+\frac{g'}{g}).$$
From the sectional curvatures, we get the scalar curvature as $$R= -2 (\epsilon \cos(r) \sinh(r) -\sin(r) \cosh(\epsilon r) +\epsilon^2 \sin(r) \cosh(\epsilon r))/fg,$$ and mean curvature of the level surface at height $$r$$ as $$\frac12(\cos(r)\cosh(\epsilon r) +\epsilon\sin(r) \sinh(\epsilon r))/fg.$$
We see that the level surfaces $$r=c$$ are mean convex tori, and the scalar curvature is positive for $$\epsilon$$ and $$r_0$$ small. Hence this metric contains no closed minimal surface: the maximal $$r$$ value for such a surface would be tangent to a level surface which is mean convex, contradicting the maximum principle.
Here's how I found this metric: given your criteria, the double of the manifold admits a metric with positive scalar curvature (this is a trick of Hubert Bray; cf. also Pengzi Miao). Such manifolds are connect sums of space forms and $$S^2\times S^1$$. The reflection symmetry quotient then gives a handlebody (a space form with non-trivial fundamental group cannot admit a reflection symmetry with fixed set a surface). Hence the simplest non-trivial case is a solid torus. The above metrics are invariant under the $$S^1\times S^1$$ action on a solid torus (a double-warped product"). The $$(r,\theta)$$ slice is a spherical cap, and the $$(r,\lambda)$$ slices are scaled hyperbolic metrics on a cyinder. Then we adjust $$\epsilon$$ and $$r_0$$ to make the scalar curvature positive, the positive sectional curvature of the spherical cap dominating the negative curvature of the hyperbolic annulus. I suspect one can use the techniques in Codá-Marques' paper to realize any handlebody with such properties.