Isolate flat umbilic on compact Riemannian surface with nonpositive curvature

Question

We know that we can have a noncompact surface with a so called Monkey saddle (https://en.wikipedia.org/wiki/Monkey_saddle), on which there is a isolated flat umbilic (where the Gaussian curvature vanishes), and the Gaussian curvature is strictly negative everywhere else.

So my question is, is there an example where there is a monkey type saddle on a COMPACT Riemannian surface with nonpositive curvature? That is, can there be an isolated flat umbilic on it? Is it possible to somehow "compactify" the above noncompact exmaple without adding positive curvature part to it?

Answer 1

Compact surfaces with non-negative Euler characteristic cannot carry such metrics because of Gauss-Bonnet. However, this is the only obstruction.

First, such metrics exist on any compact orientable surface of genus greater than 1. For example, let $M$ be a hyperelliptic Riemann surface of genus $g\ge 2$, and let $\omega_1,\ldots,\omega_g$ be a basis for the holomorphic $1$-forms on the surface. Then the Riemannian metric $$ g = \omega_1\circ\overline{\omega_1} + \cdots + \omega_g\circ\overline{\omega_g} $$ has Gaussian curvature $K$ that is negative everywhere except at the Weierstrass points (of which there are $2g{+}2$), where it vanishes.

Second, one can construct such metrics on non-orientable surfaces with negative Euler characteristic: Consider a hyperelliptic Riemann surface $S$ of genus $g>1$ that has a fixed-point-free, anti-holomorphic involution $\iota$ with the property that there exists a basis $\omega_1,\ldots,\omega_g$ as above for which $\iota^*\omega_k = \overline{\omega_k}$. (Such examples exist for each $g>1$.) Then the above metric $g$ is invariant under $\iota$ and hence descends to be a metric with the desired properties on the non-orientable surface that one obtains by quotienting $S$ by the $\mathbb{Z}_2$-action generated by $\iota$. Since this construction can be done for each $g>1$, this gives a sequence of non-orientable compact examples covering all negative Euler characteristics.

Answer 2

As a sort of illustration to the comment of Robert Bryant to his answer, - I found lots of pictures of triply periodic infinite space-filling surfaces of nonpositive curvature and discrete lattice-periodic set of zero curvature points. The requested surface must be just the quotient of this one by the three-translation group. Except that all of them are of genus $>2$. I wonder whether there is any obstacle to obtain a genus 2 surface in this way.

Notably the Wikipedia article on Triply periodic minimal surfaces contains several links to huge galleries of such images.

This is genus 3; it is nice in that eight monkey saddles are clearly visible. I wonder if this is just the metric from Robert Bryant's answer?

On John Baez's page there are even photos of 3d-printed stuff like this