Searcharxiv⌕ Search

arXiv subjects

Sebastien Picard

Publications and source records attributed to Sebastien Picard.

27 records · Page 2Linked to original sources
arXiv2018

Fu-Yau Hessian Equations

We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $α'$. Actually we obtain at the same time a solution of the open case $α'>0$, an improved solution of the known case $α'<0$, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $Γ_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.

math.DG↗
arXiv2017

The Anomaly flow over Riemann surfaces

We initiate the study of a new nonlinear parabolic equation on a Riemann surface. The evolution equation arises as a reduction of the Anomaly flow on a fibration. We obtain a criterion for long-time existence for this flow, and give a range of initial data where a singularity forms in finite time, as well as a range of initial data where the solution exists for all time. A geometric interpretation of these results is given in terms of the Anomaly flow on a Calabi-Yau threefold.

math.DG↗
arXiv2017

The Anomaly flow on unimodular Lie groups

The Hull-Strominger system for supersymmetric vacua of the heterotic string allows general unitary Hermitian connections with torsion and not just the Chern unitary connection. Solutions on unimodular Lie groups exploiting this flexibility were found by T. Fei and S.T. Yau. The Anomaly flow is a flow whose stationary points are precisely the solutions of the Hull-Strominger system. Here we examine its long-time behavior on unimodular Lie groups with general unitary Hermitian connections. We find a diverse and intricate behavior, which depends very much on the Lie group and the initial data.

math.DG↗
arXiv2017

On estimates for the Fu-Yau generalization of a Strominger system

We study an equation proposed by Fu and Yau as a natural $n$-dimensional generalization of a Strominger system that they solved in dimension $2$. It is a complex Hessian equation with right hand side depending on gradients. Building on the methods of Fu and Yau, we obtain $C^0$, $C^2$ and $C^{2,α}$ a priori estimates. We also identify difficulties in extending the Fu-Yau arguments for non-degeneracy from dimension $2$ to higher dimensions.

math.CV↗
arXiv2017

Geometric flows and Strominger systems

A geometric flow on $(2,2)$-forms is introduced which preserves the balanced condition of metrics, and whose stationary points satisfy the anomaly equation in Strominger systems. The existence of solutions for a short time is established, using Hamilton's version of the Nash-Moser implicit function theorem.

math.DG↗
arXiv2017

The Fu-Yau equation with negative slope parameter

The Fu-Yau equation is an equation introduced by J. Fu and S.T. Yau as a generalization to arbitrary dimensions of an ansatz for the Strominger system. As in the Strominger system, it depends on a slope parameter $α'$. The equation was solved in dimension $2$ by Fu and Yau in two successive papers for $α'>0$, and for $α'<0$. In the present paper, we solve the Fu-Yau equation in arbitrary dimension for $α'<0$. To our knowledge, these are the first non-trivial solutions of the Fu-Yau equation in any dimension strictly greater than $2$.

math.CV↗
arXiv2016

Concavity of the Lagrangian Phase Operator and Applications

We study the Dirichlet problem for the Lagrangian phase operator, in both the real and complex setting. Our main result states that if $Ω$ is a compact domain in $\mathbb{R}^{n}$ or $\mathbb{C}^n$, then there exists a solution to the Dirichlet problem with right-hand side $h(x)$ satisfying $|h(x)| > (n-2)\fracπ{2}$ and boundary data $ϕ$ if and only if there exists a subsolution.

math.AP↗
arXiv2013

A Priori Estimates of the Degenerate Monge-Ampere Equation on Kahler Manifolds of Nonnegative Bisectional Curvature

The regularity theory of the degenerate complex Monge-Ampère equation is studied. The equation is considered on a closed compact Kähler manifold $(M,g)$ with nonnegative orthogonal bisectional curvature of dimension $m$. Given a solution $ϕ$ of the degenerate complex Monge-Ampère equation $\det(g_{i \bar{j}} + ϕ_{i \bar{j}}) = f \det(g_{i \bar{j}})$, it is shown that the Laplacian of $ϕ$ can be controlled by a constant depending on $(M,g)$, $\sup f$, and $\inf_M Δf^{1/(m-1)}$.

math.AP↗