Positive Scalar Curvature and Lagrangian Mean Curvature Flow

Objective

The interplay between Geometry and Analysis has been among the most fruitful mathematical ideas in recent years, the most obvious example being Perelman's proof of Poincare' conjecture. The goal of this proposal is to establish a research group that will make significant progress in the following two distinct problems.

Scalar Curvature: A classical theorem in Riemannian Geometry states that nonnegative scalar curvature metrics which are flat outside a compact set must be Euclidean. The equivalent problem for positive scalar curvature is known as the Min-Oo conjecture and, after being checked in many particular cases, was recently disproven by myself and coauthors. I plan to prove optimal results relating positive scalar curvature and area of minimal surfaces. I also plan to show that these manifolds admit an infinite number of minimal surfaces (Yau's conjecture). My approach consists of studying min-max methods in order to obtain existence of higher-index minimal surfaces.

Mean curvature flow: An hard open problem consists in determining which Lagrangians in a Calabi-Yau admit a minimal Lagrangian (SLag) in their isotopy class and a complete answer would have a strong impact in Algebraic Geometry and Mirror Symmetry. A well known approach consists in deforming a given Lagrangian in the direction of its mean curvature and hope to show convergence to a SLag. The difficulty with this method is that finite-time singularities can occur.
I plan to study the regularity theory for this flow and show singularities have codimension two. This would be the ground stage to continue the flow past the singular time. My approach consists in classifying the possible parabolic blow-ups and find monotone quantities which will rule out non SLag blow-ups.

Fields of science (EuroSciVoc)

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• natural sciences mathematics pure mathematics geometry
• natural sciences mathematics pure mathematics algebra algebraic geometry

Programme(s)

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• FP7-IDEAS-ERC - Specific programme: "Ideas" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013)

Topic(s)

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• ERC-SG-PE1 - ERC Starting Grant - Mathematical foundations

Call for proposal

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ERC-2011-StG_20101014
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Funding Scheme

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ERC-SG - ERC Starting Grant

Host institution

Beneficiaries (1)