Billiards, Geometric Optics, And Geodesic Flows: Open Problems.pdf

key open problems discussed at a workshop focusing on differential geometry, billiards, and geometric optics.

Problem 1: Geodesic Flow on Cylinder-Shaped Surfaces

Context: Consider a closed, strictly convex curve (like an ellipse) bounding a domain in the plane. Imagine a surface homeomorphic to a cylinder, glued on top and bottom to this domain, where motion is along geodesics on the cylindrical part.
Question 1: Are there invariant curves (KAM curves, for example) for this geodesic flow? Are they possible near the boundary?
Question 2: What curve shapes (other than circles) make this flow integrable?
Question 3: Can the flow be ergodic?

Problem 2: Integrable Outer Billiards

Context: Outer billiards involve reflecting oriented lines after hitting a strictly convex, closed curve in the plane. The question is about finding non-elliptical curves that support integrable (constant density) outer billiards.
Question: Are there other examples of integrable outer billiards besides ellipses? This problem parallels Birkhoff's Conjecture for usual billiards.

Problem 3: Gutkin Billiards on Spheres and Hyperbolic Planes

Context: A curve with the "δ-Gutkin property" is a curve within a billiard table where incoming lines at a constant angle δ are reflected back onto the curve. Gutkin studied these in the plane; the question extends this to spheres and hyperbolic planes.
Question: What curves (not circles) in spheres and hyperbolic planes have the δ-Gutkin property?

Problem 4: Invariant Hypersurfaces in Billiards

Context: Consider a Birkhoff billiard inside a closed, strictly convex hypersurface in Euclidean space of dimension greater than 2. Lazutkin showed caustics (regions where reflections concentrate) exist for ellipsoids. The question is if other convex hypersurfaces can support invariant hypersurfaces within the billiard table.
Question: Can there be examples of billiard tables (non-ellipsoidal) with invariant hypersurfaces? What are their geometric/dynamical properties?

Problem 5: Symmetry of Caustics in Convex Billiards

Context: Consider a convex, smooth billiard table symmetric about an axis. If a caustic (a region where reflections focus) exists, is it necessarily also symmetric about the same axis?
Question: Prove or provide a counterexample to the claim that non-symmetric caustics cannot exist in this setting.

Problem 6: Projective Billiards and k-Reflectivity

Context: Projective billiards generalize usual billiards by considering bounded domains with transverse line fields on their boundaries. The question focuses on finding examples of projective billiards that are "k-reflective," meaning they have open sets of periodic points for the corresponding billiard map (for k ≥ 5 odd, currently unknown).
Question 1: Can examples of k-reflective projective billiards with k ≥ 5 odd be found?
* Question 2: Are there other examples of k-reflective projective billiards within specific classes of domains (polygonal, piecewise-algebraic, etc.)?