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  • Non-concavity of the Robin ground state
    Camb. J. Math. (IF 1.625) Pub Date : 2020-04-21
    Ben Andrews; Julie Clutterbuck; Daniel Hauer

    On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. The aim of this paper is to show that

    更新日期:2020-04-21
  • Existence of hypersurfaces with prescribed mean curvature I – generic min-max
    Camb. J. Math. (IF 1.625) Pub Date : 2020-04-21
    Xin Zhou; Jonathan J. Zhu

    We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersurface of multiplicity one. More precisely, we show that our previous min-max theory

    更新日期:2020-04-21
  • The index and nullity of the Lawson surfaces $\xi_{g,1}$
    Camb. J. Math. (IF 1.625) Pub Date : 2020-04-21
    Nikolaos Kapouleas; David Wiygul

    We prove that the Lawson surface $\xi_{g,1}$ in Lawson’s original notation, which has genus $g$ and can be viewed as a desingularization of two orthogonal great two-spheres in the round three-sphere $\mathbb{S}^3$, has index $2g + 3$ and nullity $6$ for any genus $g \geq 2$. In particular $\xi_{g,1}$ has no exceptional Jacobi fields, which means that it cannot “flap its wings” at the linearized level

    更新日期:2020-04-21
  • $(1,1)$ forms with specified Lagrangian phase: a priori estimates and algebraic obstructions
    Camb. J. Math. (IF 1.625) Pub Date : 2020-04-21
    Tristan C. Collins; Adam Jacob; Shing-Tung Yau

    Let $(X, \alpha)$ be a Kähler manifold of dimension $n$, and let $[\omega] \in H^{1,1} (X, \mathbb{R})$. We study the problem of specifying the Lagrangian phase of $\omega$ with respect to $\alpha$, which is described by the nonlinear elliptic equation\[\sum^{n}_{i=1} \arctan (\lambda_i) = h(x)\]where $\lambda_i$ are the eigenvalues of $\omega$ with respect to $\alpha$. When $h(x)$ is a topological

    更新日期:2020-04-21
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