Mean Curvature Problems in Differential Geometry

The study of mean curvature occupies a central place in modern differential geometry, linking geometric analysis, partial differential equations and applications across physics and materials science. Mean curvature is defined as the divergence of the unit normal vector field on a hypersurface, encapsulating the notion of how rapidly a surface bends within an ambient space. Problems involving prescribed mean curvature equations arise in the characterisation of minimal surfaces, capillary interfaces and steady-state configurations in general relativity. Such equations are quasilinear elliptic or hyperbolic systems, often formulated in Euclidean or Lorentzian settings, and pose challenges owing to their non-uniform ellipticity and nonlinear boundary conditions. Research has focused on existence and uniqueness of solutions, multiplicity and bifurcation phenomena, and qualitative properties such as tangential regularity or singularity formation. Techniques range from variational methods and geometric measure theory to topological degree theory and fixed-point arguments. In the Minkowski-space context, one studies non-potential and non-radial Dirichlet systems for the mean curvature operator, revealing intricate parameter regimes for existence and multiplicity. Advances in this field not only deepen the theoretical understanding of geometric flows and analytic curvature operators but also inform practical models for fluid membranes, material interfaces and relativistic membranes.

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Mean Curvature Problems in Differential Geometry publication trend

The graph below shows the total number of articles in mean curvature problems in differential geometry across all publications each year (not limited to Nature Index journals).

Technical terms

Mean curvature: The average of principal curvatures of a surface, given by the divergence of its unit normal.

Mean curvature operator: A quasilinear differential operator expressing divergence of the normalised gradient in Euclidean or Minkowski space.

Dirichlet problem: A boundary value problem where the solution is prescribed on the boundary of the domain.

Bifurcation curve: A locus in parameter space at which the number or stability of solutions changes qualitatively.

Minkowski space: A Lorentzian manifold with a metric of signature (−,+,+,…) used in special relativity and relativistic curvature models.

References

1. Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space. Discrete and Continuous Dynamical Systems (2020).
2. π/4-tangentiality of solutions for one-dimensional Minkowski-curvature problems. Advances in Nonlinear Analysis (2020).
3. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete and Continuous Dynamical Systems (2019).