Singularities of axially symmetric time-like minimal submanifolds in Minkowski space

We prove that there does not exist global-in-time axisymmetric solutions to the time-like minimal submanifold system in Minkowski space. We further analyze the limiting geometry as the maximal time of existence is approached.

PINCHING THEOREMS FOR TOTALLY REAL MINIMAL SUBMANIFOLDS INCPn

LetMbe ann-dimensional totally real minimal submanifold inCPn. We prove that ifMis semi-parallel and the scalar curvature τ,$\frac{-(n-1)(n-2)(n+1)}{2}\leq \tau \leq 0$, thenMis an open part of the Clifford torusTn⊂CPn. IfMis semi-parallel and the scalar curvature τ,$n(n-1)\leq \tau \leq \frac{n^{3}-3n+2}{2}$, thenMis an open part of the real projective spaceRPn.

PINCHING THEOREMS FOR A COMPACT MINIMAL SUBMANIFOLD IN A COMPLEX PROJECTIVE SPACE

We give a formula for the Laplacian of the second fundamental form of an n-dimensional compact minimal submanifold M in a complex projective space CPm. As an application of this formula, we prove that M is a geodesic minimal hypersphere in CPm if the sectional curvature satisfies K≥1/n, if the normal connection is flat, and if M satisfies an additional condition which is automatically satisfied when M is a CR submanifold. We also prove that M is the complex projective space CPn/2 if K≥3/n, and if the normal connection of M is semi-flat.

On Lagrangian Catenoids

Recently I. Castro and F.Urbano introduced the Lagrangian catenoid. Topologically, it is ℝ × Sn–1 and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ℂn is foliated by round (n – 1)-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ℂn. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid.

An application of Moser iteration to complete minimal submanifolds in a sphere

We apply the Moser iteration method to obtain a pointwise bound on the norm of the second fundamental form from a bound on its Ln norm for a complete minimal submanifold in a sphere. As an application we show that a complete minimal submanifold in a sphere with finite total curvature and Ricci curvature bounded away from -∞ must be compact. This complements similar results of Osserman and Oliveira in the case the ambient space is the Euclidean and the hyperbolic space respectively.

Three theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of the Cayley algebra

It is proved that cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of the octave algebra are ruled manifolds. A necessary and sufficient condition for a cosymplectic hypersurface of a Hermitian submanifoldM6⊂Oto be a minimal submanifold ofM6is established. It is also proved that a six-dimensional Hermitian submanifoldM6⊂Osatisfying theg-cosymplectic hypersurfaces axiom is a Kählerian manifold.