Harmonic Function with special property

Question

I would appreciate any help with the following problem:

Let $(M,g)$ be a 3 dimensional Riemann manifold with boundary. Let $ \Gamma $ be a surface of sufficient regularity dividing M into two connected components $M_{+}$ and $M_{-}$. Is it possible to construct a harmonic function such that it is zero on $\Gamma$ and it is non-zero away from it everywhere in the manifold? As a follow up question is it possible to construct a harmonic function which is zero on $\Gamma$ and monotone in normal direction to $\Gamma$?

Two comments are due: it is clear that by setting $v=0$ on $\Gamma$ and $ dv \neq 0$ we can achieve the first question locally but not globally in M. second comment: one can use maximum principle in following way; put dirichlet data 1 on $\partial M_{+}$ and zero on $\Gamma$. put Dirichlet data -1 on $\partial M_{-}$ in this way we are guaranteed that the harmonic function would never be zero away from the surface but the first normal derivatives on $\Gamma$ will not necessarily match.

cheers,

Answer 1

Depends on what you mean by "sufficient regularity". No simple interpretation will work. Think of the disk $|z|<10$ on the plane split by the parabola $y=x^2$. Put $v(z)=u(z+iz^2)$. Then $v$ is still harmonic in the disk $|z|<2$ and vanishes on its real diameter $(-2,2)$. Thus we have the identity $v(\bar z)=-v(z)$ by the Schwartz reflection principle, so $u(\bar z+i\bar z^2)=-u(z+i z^2)$ for $|z|<2$. Taking $z=i$, we get $u(-2i)=-u(0)=0$ and up the chimney go all our hopes for good life. In a sense, the behavior of the Schwartz reflection about $\Gamma$ on $M$ is responsible for everything here in all cases but I wish you good luck expressing the corresponding conditions in human terms...

Answer 2

Despite @fedja's trenchant observations/example, there are configurations that are not hostile to a construction as suggested. In 2D, for example, on an annulus (with concentric circles), it is easy to make harmonic functions that are 0 on one of the bounding circles but not on the other.

In 3D (and generally), a similar thing can be accomplished, although with less commonly-understood notations or conventions, on concentric spherical shells.

(We note that there is no Riemann mapping theorem in higher dimensions, so the 2D examples' quasi-universality fades in their higher dimensional analogues...)

Answer 3

Despite the interesting observation above, can something be said in the three dimensional case as that is what I am mostly interested in. I can even rephrase the question to be more rigorous as follows:

given a minimal surface $\Gamma$ in a 3-dimensional riemannian manifold does there exist a harmonic function whose zero level set is $\Gamma$?