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## LAPLACIAN OF HYPERSURFACE OF A SPHERE

If $M$ is a connected hypersurface of $S^{d+1}$ closed dividing the sphere into two connected components, and if $u$ is a function on the ambient sphere whose restriction to $M$ is $f$ then:

$\Delta_{S^{d+1}} u(x) = (N^2 + d H N)(u) + \Delta_M f(x)$

for points $x\in M$ where $N$ is the unit normal on $M$. See Pliakis 2014.

Now let’s try to understand what the Schroedinger equation:

$i \hbar \frac{d}{dt} \psi = \hbar^2 (\Delta_M + V)\psi$

can mean slowly and carefully.  Let’s totally open up our imagination to consider the possibility that this potential $V$ is exactly something like $(N^2 + d H)$ because that is very interesting.