arXiv subjects
Publications and source records attributed to Samuel Sottile.
We prove a Weyl-type theorem for the Kohn Laplacian on sphere quotients as CR manifolds. We show that we can determine the fundamental group from the spectrum of the Kohn Laplacian in dimension three. Furthermore, we prove Sobolev estimates for the complex Green's operator on these quotient manifolds.
We obtain an analog of Weyl's law for the Kohn Laplacian on lens spaces. We also show that two 3-dimensional lens spaces with fundamental groups of equal prime order are isospectral with respect to the Kohn Laplacian if and only if they are CR isometric.
This work introduces the Hookean-Voronoi energy, a minimal model for the packing of soft, deformable balls. This is motivated by recent studies of quasi-periodic equilibria arising from dense packings of diblock and star polymers. Restricting to the planar case, we investigate the equilibrium packings of identical, deformable objects whose shapes are determined by an $N$-site Voronoi tessellation of a periodic rectangle. We derive a reduced formulation of the system showing at equilibria each site must reside at the ``max-center'' of its associated Voronoi region and construct a family of ordered ``single-string'' minimizers whose cardinality is $O(N^2)$. We identify sharp conditions under which the system admits a regular hexagonal tessellation and establish that in all cases the average energy per site is bounded below by that of a regular hexagon of unit size. However, numerical investigation of gradient flow of random initial data, reveals that for modest values of $N$ the system preponderantly equilibrates to quasi-ordered states with low energy and large basins of attraction. For larger $N$ the distribution of equilibria energies appears to approach a $\delta$-function limit, whose energy is significantly higher than the ground state hexagon. This limit is possibly shaped by two mechanisms: a proliferation of moderate-energy disordered equilibria that block access of the gradient flow to lower energy quasi-ordered states and a rigid threshold on the maximum energy of stable states.
We study the geometry of geodesics on $\mathsf{SL}(n)$, equipped with the Hilbert-Schmidt metric which makes it a Riemannian manifold. These geodesics are known to be related to affine motions of incompressible ideal fluids. The $n = 2$ case is special and completely integrable, and the geodesic motion was completely described by Roberts, Shkoller, and Sideris; when $n = 3$, Sideris demonstrated some interesting features of the dynamics and analyzed several classes of explicit solutions. Our analysis shows that the geodesics in higher dimensions exhibit much more complex dynamics. We generalize the Virial-identity-based criterion of unboundedness of geodesic given by Sideris, and use it to give an alternative proof of the classification of geodesics in 2D obtained by Roberts--Shkoller--Sideris. We study several explicit families of solutions in general dimensions that generalize those found by Sideris in 3D. We additionally classify all "exponential type" geodesics, and use it to demonstrate the existence of purely rotational solutions whenever $n$ is even. Finally, we study "block diagonal" solutions using a new formulation of the geodesic equation in first order form, that isolates the conserved angular momenta. This reveals the existence of a rich family of bounded geodesic motions in even dimension $n \geq 4$. This in particular allows us to conclude that the generalization of the swirling and shear flows of Sideris to even dimensions $n \geq 4$ are in fact dynamically unstable.