Minimal Surfaces (GMT Perspective)¶

Job: state the Plateau problem and the existence/regularity package for area-minimizing surfaces in the language of currents, varifolds, and sets of finite perimeter; contrast with classical differential geometry.

Core¶

Classical vs weak notions¶

A smooth oriented hypersurface $M\subset\mathbb{R}^{n}$ (or Riemannian ambient) is minimal if mean curvature $H\equiv 0$ — critical point of area for compactly supported variations. Stable: second variation $\delta^2\mathrm{Area}\ge 0$. Area-minimizing: globally minimizes area among competitors with the same boundary (or in a homology class).

Category	Object	First variation	Boundary
Classical DG	Immersed submanifold	$H=0$	prescribed smooth $\Gamma$
Finite perimeter	Caccioppoli set $E$	$H$ measure on $\partial^*E$	$\partial^*E$ free or partial
Integral currents	$T\in\mathbf{I}_k$	stationary / minimizing mass	$\partial T$ fixed
Varifolds	$V$ on Grassmannian	stationary ($\delta V=0$)	free boundary variants

Plateau problem¶

Classical Plateau (Douglas–Radó, $k=2$ in $\mathbb{R}^3$). Given a closed Jordan curve $\Gamma\subset\mathbb{R}^3$, there exists a branched minimal immersion of the disk with boundary $\Gamma$ minimizing the Dirichlet energy / area in the parametrized class.

GMT Plateau (Federer–Fleming). Given an integer multiplicity boundary $S=\partial T_0$ with finite mass, there exists an integral current $T$ with $\partial T=S$ minimizing $\mathbf{M}(T)$. Proof schema: minimizing sequence $\to$ weak limit by compactness; lower semicontinuity of mass; Federer–Fleming deformation / closure theorem keeps the limit integral.

Oriented least area in a homology class: same package on a compact Riemannian manifold — existence of mass-minimizing integral currents in a given class.

Regularity theory (one-liners)¶

Interior regularity for mass-minimizing hypersurfaces (integral currents / perimeter minimizers).

$n\le 7$: area-minimizing hypersurfaces in $\mathbb{R}^n$ (boundaries of least perimeter sets) are smooth (actually analytic if ambient analytic). Simons: no singular stable minimal cones through dimension $6$ in the ambient Euclidean dimension $\le 7$.
$n\ge 8$: singular set allowed; Federer dimension reduction $\Rightarrow$ $\dim_{\mathcal{H}}\operatorname{sing}T\le n-8$ for minimizing hypersurfaces. Simons cone in $\mathbb{R}^8$ is area-minimizing and singular at the origin (Bombieri–De Giorgi–Giusti).
Higher codimension: Almgren’s big regularity theorem — singular set of $m$-dimensional mass-minimizing currents in $\mathbb{R}^n$ has dimension $\le m-2$; much harder (Q-valued functions).

Allard regularity (stationary varifolds). If an integral stationary varifold is weakly close to a plane with small mass excess in a ball, then it is $C^{1,\alpha}$ graphical in a smaller ball. Threshold for “almost minimizing / small excess $\Rightarrow$ smooth.”

Schoen–Simon–Yau / Schoen–Simon. Curvature estimates and regularity for stable hypersurfaces under ambient curvature and volume growth hypotheses — bridge between stability and GMT compactness.

Boundary regularity. Hardt–Simon: area-minimizing hypersurfaces with smooth boundary data are smooth up to the boundary in low dimensions; singular set still dimension-controlled.

GMT vs classical differential geometry¶

Aspect	Classical DG	GMT
Objects	Smooth (or $C^2$) immersions / embeddings	Rectifiable sets, currents, varifolds
Existence	Often needs a priori estimates / continuity method	Direct method: compactness + l.s.c.
Singularities	Often excluded by assumption	Controlled measure-theoretic singular sets
Calculus	PDE for $H$, Jacobi operator	First/second variation of mass; monotonicity formulae
Topology	Fixed genus / parametrization (Douglas–Radó)	Homological boundaries; topology recovered a posteriori
Strength	Fine structure (index, Morse theory, CMC)	Global existence; weak limits; large-scale geometry

Monotonicity formula (common core). For stationary surfaces, $r\mapsto r^{-k}\mathcal{H}^k(M\cap B_r(x))$ is monotone — forces conical blow-ups and feeds dimension reduction.

Mean curvature flow (pointer). Brakke’s weak MCF uses varifolds; Huisken–White regularity vs singularities complements the elliptic minimal surface theory.

See it¶

Simons cone. $$ C=\bigl{(x,y)\in\mathbb{R}^4\times\mathbb{R}^4:|x|=|y|\bigr}\subset\mathbb{R}^8 $$ is a minimal cone, stable, and area-minimizing (BDG). It is singular only at $0$, matching the $n-8=0$ dimensional singular set bound in ambient dimension $8$.

Isoperimetric problem as perimeter minimization. Minimize $P(E)$ subject to $|E|=c$ $\Leftrightarrow$ constant mean curvature reduced boundary (Lagrange multiplier); balls are the unique minimizers in $\mathbb{R}^n$ (up to null sets).

Check¶

Why does the direct method fail in the class of smooth surfaces with fixed boundary, and how do currents repair it?

Smooth area-minimizing sequences may develop necks, multiplicity, or lose topology; $C^\infty$ is not compact under area bounds. Integral currents add weak topology with mass-compactness and still encode orientation and boundary; regularity theory recovers smoothness a.e. afterwards.

State the optimal Hausdorff dimension bound for the interior singular set of a mass-minimizing hypersurface in $\mathbb{R}^n$.

$\dim_{\mathcal{H}}\operatorname{sing}\le n-8$ (empty for $n\le 7$). Sharp by the Simons cone and products with Euclidean spaces.

Pitfalls¶

Stationary $\neq$ minimizing: stationary varifolds can be unstable or not globally minimizing; regularity theory differs in strength.
Multiplicity: minimizing currents may carry multiplicity; smooth embeddedness is extra (e.g. around boundary or under topological constraints).
Codimension $>1$: branch points and Almgren’s $m-2$ bound — do not quote the $n-8$ hypersurface theorem.
Parametrized Plateau (maps from surfaces) vs currents: Douglas–Radó solutions can branch; GMT solutions minimize mass but may a priori have uncontrolled topology until regularity/structural results apply.

Refs¶

Giusti, Minimal Surfaces and Functions of Bounded Variation
Simon, Lectures on Geometric Measure Theory
Colding–Minicozzi, A Course in Minimal Surfaces (classical + modern DG)
Morgan, Geometric Measure Theory: A Beginner’s Guide
Federer, Geometric Measure Theory
Almgren, Almgren’s Big Regularity Paper (Q-valued functions)

Next¶

GMT overview · Foundations · Differential geometry · Measure theory · PDE foundations · Calculus of Variations — minimal surfaces as variational minimizers