The Height Of Compact Nonsingular Heisenberg-like Nilmanifolds

We consider nonsingular Heisenberg-like nilmanifolds. This class
strictly contains the well-known Heisenberg-type nilmanifolds. For any

compact nonsingular Heisenberg-like nilmanifold, we present a trace
formula. This formula enables us to access the height, i.e., the
derivative of the spectral zeta function at zero.

We then investigate the global behaviour of the height as a function on
the moduli space of volume-normalised Heisenberg-like metrics on a fixed
compact nonsingular nilmanifold. If the dimension of the underlying
manifold is 3 mod 4 then the height attains a global minimum. In
contrast, one can construct paths of metrics on which the height is not
bounded from below if the dimension is 0 or 1 mod 4. The existence of
global minima can be enforced in any dimension by bounding the sectional
curvature from above.

At last, we will construct the global minimiser of the height in
dimension 3 and consider certain local minima of Heisenberg-type metrics
in dimensions 5, 9 and 25.