Open Problems in Geometry of Curves and Surfaces

Though largely resolved for tori, the general Willmore conjecture remains open for surfaces of higher genus. The conjecture asks for the surface of genus g that minimizes the Willmore energy, defined as (k-k)dA, where k and k are the principal curvatures. While Marques and Neves proved the case for tori in 2012, determining the minimizers for surfaces of genus g 2 remains an active area of research.

The classification of constant mean curvature (CMC) surfaces in three-dimensional Riemannian manifolds presents significant challenges. While Delaunay surfaces completely characterize CMC surfaces in Euclidean space, finding complete characterizations in other geometric settings, particularly spaces of non-constant curvature, remains elusive. The Alexandrov theorem states that compact embedded CMC surfaces in Euclidean space must be spheres, but analogous results in other settings are incomplete.

This problem seeks a complete characterization of metrics conformal to the standard metric on the sphere that have constant scalar curvature. Posed by Louis Nirenberg in 1974, it asks which functions on the sphere can be the Gauss curvature of a metric conformal to the standard one. While solved in special cases, the general problem connects to the study of nonlinear elliptic partial differential equations and variational methods on manifolds, with connections to the Yamabe problem.

While John Nash proved that any smooth manifold can be isometrically embedded in Euclidean space, determining the optimal embedding dimensions for various classes of manifolds remains open. Specifically, finding the minimal dimension m such that every n-dimensional Riemannian manifold can be isometrically embedded in R^m is unresolved for many specific classes of manifolds. The current general bound of m = n(3n+11)/2 is likely not optimal.

Hilbert's 16th problem seeks a complete topological classification of real algebraic curves. While Harnack's inequality provides an upper bound on the number of components a real algebraic curve of degree d can have, determining exactly which configurations within this bound are possible remains an open challenge. The problem becomes particularly complex for curves of high degree and surfaces of higher dimension.

Understanding which varieties are rational (birationally equivalent to projective space) remains a central challenge. While rationality criteria are well-understood for curves and low-dimensional varieties, determining rationality for higher-dimensional algebraic varieties presents significant difficulties. The nonrationality of cubic hypersurfaces was established only recently, but the general classification of rational varieties remains incomplete.

Developing provably correct algorithms that can reconstruct smooth curves and surfaces from discrete point samples, especially in the presence of noise, remains challenging. The problem is to determine under what sampling conditions and with what computational complexity one can reconstruct a curve/surface that faithfully represents the original shape. This has applications in medical imaging, computer vision, and many other fields.

Creating numerical meshes that accurately capture the geometry of curved surfaces while maintaining computational efficiency presents ongoing challenges. Issues involving anisotropic meshes, adaptive refinement for regions of high curvature, and handling surfaces with non-trivial topology remain areas of active research. Theoretical guarantees on mesh quality and computational complexity for complex curved geometries are incomplete.

The question of classifying minimal surfaces in arbitrary three-dimensional manifolds remains largely open. While the geometry of minimal surfaces in space forms (spheres, Euclidean space, hyperbolic space) is relatively well-understood, extending these results to manifolds with more complex geometry presents significant challenges. Understanding the existence, uniqueness, and stability of minimal surfaces in general three-manifolds connects to the study of geometric variational problems.

The curve shortening flow, which evolves a curve according to its curvature, presents several open problems regarding singularity formation. While Grayson proved that any embedded curve in the plane shrinks to a point without developing singularities, understanding the behavior of curves on surfaces, particularly those with non-positive curvature, remains problematic. The classification of singularities that can form during this flow in more general settings is incomplete.

Optimal systolic inequalities, which relate the size of a manifold to the lengths of its shortest non-contractible curves, remain unknown for many classes of surfaces. While Loewner's torus inequality and Pu's theorem for the real projective plane provide important results, establishing sharp bounds for surfaces of higher genus and finding which metrics realize these bounds present ongoing challenges.