Spectral Geometry and its Relations to Minimal Surfaces

Copenhagen Centre for Geometry & Topology (GeoTop) at the University of Copenhagen
August 17-21 2026, 


The aim of this masterclass is to explore the interplay between Spectral Geometry and Minimal Surface theory.  Minimal surfaces, one of the oldest topics in geometry, concern special surfaces that are locally area-minimizing. Spectral geometry, famously popularized by Kac’s famous question “Can one hear the shape of a drum?”, studies the eigenfunctions and eigenvalues associated with differential operators on manifolds.  This masterclass is designed to introduce the exciting interactions between these two areas, present recent developments, and highlight open problems and directions for future research.

 The course is intended for MSc students, PhD candidates, and postdoctoral researchers working in differential geometry, geometric analysis, PDE, Spectral theory, and related areas. 

Limited funding will be available for participants.

Click here to see a bigger version of the poster

 

Mikhail Karpukhin (UCL ᴜᴋ)  

Title: Equivariant minimal surfaces and spectral geometry

The study of sharp upper bounds for Laplacian eigenvalues under area constraints is a classical topic in spectral geometry. A key source of interest lies in the remarkable fact that metrics achieving equality in such bounds correspond to metrics induced by minimal surfaces in spheres. A similar correspondence exists between optimisers of Steklov eigenvalues and free boundary minimal surfaces in the ball. These connections become especially powerful in the presence of a rich finite group of isometries. In this series of lecture we will discuss recent advances in the field such as (1) constructions of new families of minimal surfaces in low-dimensional targets; (2) estimates on the index of minimal surfaces and (3) uniqueness of minimal surfaces under symmetry assumptions. The course will focus on applying the ideas from spectral theory to classical problems in geometric analysis.

Vanderson Lima (UNB ʙʀ)

Title: Eigenvalue problems and free boundary minimal surfaces in spherical caps

Class 1- Free boundary minimal surfaces in geodesic balls of space forms
We will discuss some classical results about free boundary minimal surfaces in geodesic balls of space forms. Moreover, we will also present some examples of such objects and discuss how they lead to the Steklov problem with frequency.
Class 2 - The Steklov eigenvalue problem with frequency
We will define the Dirichlet-to-Neumann operator with frequency on a compact manifold with boundary and study the basic properties of its spectrum.
Class 3 - Free boundary minimal surfaces in spherical caps via shape optimization 
For a compact orientable surface with boundary, we define a functional on an open set of the space of its Riemannian metrics, and characterize the critical points as certain free boundary minimal surfaces.
Class 4 - Uniqueness of free boundary minimal annuli in spherical caps
We will present the family of rotational free-boundary minimal annuli in spherical caps and prove a characterization result for this family in terms of eigenfunctions of the Steklov problem with frequency.
Class 5 - Further results and open problems
We will discuss how the theory of free boundary minimal surfaces in geodesic balls of space forms has evolved in the last years and present some open problems.

Romain Petrides (IMJ-PRG ꜰʀ)

Title: Variational methods on eigenvalue functionals and applications

In this series of lectures, we present variational methods for proving the existence and non-existence of extremal metrics for eigenvalue functionals, as well as for more general functionals involving combinations of eigenvalues. The approach is based on the characterisation of extremal metrics as solutions to geometric nonlinear partial differential equations, including - but not limited to - equations for minimal surfaces.
Throughout the course, we will illustrate these methods through examples drawn from two main areas of application: (1) the computation of sharp geometric bounds for eigenvalue problems; (2) the construction of solutions to geometric nonlinear partial differential equations by eigenvalue optimisation.

 

 

All lectures will take place in Aud 10, H.C. Ørsted Building, Universitetsparken 5. (Google map link to the location of the masterclass)

Click here to see the schedule

 

 

The conference/masterclass will take place at the Department of Mathematical Sciences, University of Copenhagen. See detailed instructions on how to reach Copenhagen and the conference venue.

Tickets and passes for public transportation can be bought at the Copenhagen Airport and every train or metro station. You can find the DSB ticket office on your right-hand side as soon as you come out of the arrival area of the airport. DSB has an agreement with 7-Eleven, so many of their shops double as selling points for public transportation.

A journey planner in English is available.

More information on the "find us" webpage.

 

 

Registration is closed. 

 

 

 

 

 

 

 

 

 

 

Priya Kaveri, pkvv@math.ku.dk 

Niels Martin Møller, nmoller@math.ku.dk 

Administrator: Jan Tapdrup, jt@math.ku.dk