Geometric Hydrodynamics in Open Problems — Problem 31

Definitions and mathematical statement

Let $V>0$ denote the fixed enclosed volume. Symbolically, the shape space under consideration is \[ \mathcal S_V(M)=\{\Sigma=\partial\Omega\subset M:\ \Sigma\text{ is a closed hypersurface and }\operatorname{Vol}_M(\Omega)=V\}, \] where shapes are considered independently of their parametrizations. The ambient space $M$, its geometry, the admissible regions $\Omega$, and the regularity of their boundaries are not specified in the supplied record; this notation records the intended objects without selecting those missing data. The source also describes these objects through constant densities supported inside the hypersurfaces, without specifying a density normalization.

In a smooth Riemannian realization, a normal velocity $u$ preserving enclosed volume satisfies \[ \int_\Sigma u\,{\,\mathrm{d}} A=0. \] This is a conditional description of the volume constraint, not an additional ambient hypothesis supplied by the source. The notation $H^{-1/2}(\Sigma)$ denotes the negative Sobolev space dual to $H^{1/2}(\Sigma)$, using the surface pairing when that realization is available.

The source considers nonlocal metrics $g$ of Sobolev order $-1/2$ on these shape spaces, constructed using Neumann-to-Dirichlet operators. Such operators send prescribed normal derivatives of harmonic potentials to their boundary values, subject to the compatibility conditions and normalizations of the relevant boundary problem. The source does not specify those boundary problems, the operator combination defining $g$, or its normalization. Sobolev order alone therefore does not determine a particular metric.

Problem (Khesin, Misiolek, Modin, 2022, Problem 31). Describe the differential geometry of shape spaces equipped with these $H^{-1/2}$-metrics arising from dynamic fluid boundaries or vortex sheets.

The supplied motivation states that these metrics are nondegenerate because they are bounded below by the Kantorovich--Wasserstein distance, and that their geodesics describe potential fluid flows with vortex sheets and fluids with free dynamic boundary. The precise distance comparison, density normalization, fluid domains, and boundary conditions are not supplied. These are retained as source assertions requiring clarification, rather than formulated here as additional theorems. The request to describe the differential geometry is an open-ended research task; the record specifies neither particular geometric invariants to determine nor a single conjectural conclusion.

Short English statement

Describe the differential geometry of spaces of closed hypersurfaces enclosing a fixed volume, equipped with the nonlocal $H^{-1/2}$-metrics obtained from fluid boundary or vortex-sheet constructions using Neumann-to-Dirichlet operators. The supplied record leaves the ambient setting and exact metric construction unspecified and labels the problem open.

Sources

• Local UnsolvedMath registry, permanent ID $800832$, source problem code catalogue entry, titled Geometric Hydrodynamics in Open Problems --- Problem 31. The registry attributes the problem to Khesin, Misiolek, Modin (2022) and supplies the status open; this status is not independently verified here.
• Khesin, Misiolek, Modin, Geometric Hydrodynamics in Open Problems (2022), Problem 31. Supplied source URL: https://arxiv.org/abs/2205.01143. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of access during this drafting.
• The source invokes reference [IK18] for the metric construction, and references [IK18, Marsden, Loe] for its hydrodynamical interpretation. Bibliographic entries and links for these reference labels are not included in the supplied record, so their identities are not expanded here.