A minimal submanifold of a Riemannian manifold is an immersed submanifold whose mean curvature vector vanishes. It is a stationary point of volume under compactly supported variations of the immersion. Stationarity does not require local or global volume minimization. A minimal surface is the two-dimensional hypersurface case in a three-dimensional ambient manifold. Totally geodesic submanifolds are minimal, while the minimal Clifford hypersurfaces in a round sphere have nonzero second fundamental form and are also minimal. The Solomon-Yau least-volume conjecture compares closed minimal hypersurfaces in spheres with these products.
Existence theorems for two-dimensional minimal submanifolds can also prescribe finite symmetries. Let 
be the interior of a connected compact domain with nonempty smooth boundary
in a Riemann surface. Let a finite
group 
act faithfully by holomorphic automorphisms
on 
, with an extension to its closure that acts freely
on the boundary. Suppose 
also acts by orthogonal
transformations on 
, where 
. Assume that, for each nontrivial stabilizer 
, there is a conformal 
-equivariant map 
that is an injective linear
transformation, where 
is the tangent space.
Alarcón and Forstnerič (2026) proved the existence of a conformal immersion 
that is minimal and is both a 
-equivariant map and a proper map. Under the same hypotheses, Alarcón
and Forstnerič (2026) constructed a complete conformal minimal immersion 
that is a 
-equivariant map and a proper map into the open unit
ball 
. Completeness concerns the induced Riemannian
metric.
A minimal hypersurface is the codimension-one case.
