Strict hot spots on simply connected planar domains
On a bounded simply connected planar domain with smooth boundary, no eigenfunction of the first positive Neumann eigenvalue has an interior critical point, so its maximum and minimum are attained only on the boundary: Burdzy's simply connected hot-spots conjecture, in strict form and for every eigenfunction. The proof: Filonov's gap between the Neumann and Dirichlet eigenvalues; a conformal map smooth up to the boundary, boundary signs and the multiplicity bound; Rohleder's variational principle for tangent vector fields; boundary kernels of the disc with a subcritical potential and a multiplier identity; conditional negativity of a reciprocal kernel through matrices with one positive eigenvalue; a Hilbert-space embedding of the circle. Standard facts on Sobolev spaces and elliptic regularity are named hypotheses. The course is an exposition of OpenAI's proof (2026).
Six lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Neumann eigenfunctions and the hot-spots problem
- Conformal coordinates, Hopf's lemma and boundary signs
- A variational principle for tangent vector fields
- Boundary kernels and the multiplier identity
- Reciprocal kernels and conditional negativity
- The absence of interior critical points
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Dirichlet series and Euler products · The Riemann zeta function
- Anisotropic derivative classes and analyticity (Constant-coefficient equations and solvability)
- Boundary energy, local inverses, and harmonic data
- Hilbert spaces and compact operators · Foundations of von Neumann algebras
- Weak solutions and boundary regularity of the Dirichlet problem (course Elliptic operators and boundary problems); analyticity of solutions of elliptic equations with constant coefficients (course Constant-coefficient equations and solvability); Cauchy's theorem and the harmonic maximum principle (course The Riemann zeta function, first lesson, Appendix A); Hilbert spaces (course Foundations of von Neumann algebras).