arXiv subjects
Publications and source records attributed to Adam Jacob.
We establish a convergence result for the mean curvature flow starting from a totally real submanifold which is "almost minimal" in a precise, quantitative sense. This extends, and makes effective, a result of H. Li for the Lagrangian mean curvature flow.
The line bundle mean curvature flow is a complex analogue of the mean curvature flow for Lagrangian graphs, with fixed points solving the deformed Hermitian-Yang-Mills equation. In this paper we construct two distinct examples of singularities along the flow. First, we find a finite time singularity, ruling out long time existence of the flow in general. Next we show long time existence of the flow with a Calabi symmetry assumption on the blowup of $\mathbb P^n$, $n\geq 3$, if one assumes supercritical phase. Using this, we find an example where a singularity occurs at infinite time along the destabilizing subvariety in the semi-stable case.
Suppose $v(x,y):\mathbb C\rightarrow \mathbb R$ is an entire harmonic polynomial with no critical points in the right half plane. Let $z_1, z_2\in\mathbb C$ lie on a level set of $v$ , and assume ${\rm Re}(z_2)>{\rm Re}(z_1)\geq0$. We give a necessary and sufficient condition, depending only on algebraic properties of the polynomial $v$, for when there exists a smooth real function $f$ whose graph $x+if(x)$ lies on a level curve of $v$ connecting $z_1$ to $z_2$. Inspired by GIT, we construct a Kempf-Ness functional on an appropriate function space, and prove the functional is bounded from below and proper if and only if a such a graph exists. As an application, we find a stability condition equivalent to the existence of a solution to the deformed Hermitian-Yang-Mills equation on the family of projective bundles $ X_{r,m}:=\mathbb P(\mathcal O_{\mathbb P^m}\oplus \mathcal O_{\mathbb P^m}(-1)^{\oplus (r+1)})$ with Calabi Symmetry.
We give a short proof of the Torelli theorem for $ALH^*$ gravitational instantons, using the authors' previous construction of mirror special Lagrangian fibrations in del Pezzo surfaces and rational elliptic surfaces together with recent work of Sun-Zhang. In particular, this includes an identification of 10 diffeomorphism types of $ALH^*_b$ gravitational instantons.
We prove a version of the Strominger-Yau-Zaslow mirror symmetry conjecture for non-compact Calabi-Yau surfaces arising from, on the one hand, pairs $(\check{Y},\check{D})$ of a del Pezzo surface $\check{Y}$ and $\check{D}$ a smooth anti-canonical divisor and, on the other hand, pairs $(Y,D)$ of a rational elliptic surface $Y$, and $D$ a singular fiber of Kodaira type $I_k$. Three main results are established concerning the latter pairs $(Y,D)$. First, adapting work of Hein \cite{Hein}, we prove the existence of a complete Calabi-Yau metric on $Y\setminus D$ asymptotic to a (generically non-standard) semi-flat metric in every K\"ahler class. Secondly, we prove a uniqueness theorem to the effect that, modulo automorphisms, every K\"ahler class on $Y\setminus D$ admits a unique asymptotically semi-flat Calabi-Yau metric. This result yields a finite dimensional K\"ahler moduli space of Calabi-Yau metrics on $Y\setminus D$. Further, this result answers, in this setting, questions of Tian-Yau and Yau. Thirdly, building on the authors' previous work, we prove that $Y\setminus D$ equipped with an asymptotically semi-flat Calabi-Yau metric $\omega_{CY}$ admits a special Lagrangian fibration whenever the de Rham cohomology class of $\omega_{CY}$ is not topologically obstructed. Combining these results we define a mirror map from the moduli space of del Pezzo pairs $(\check{Y}, \check{D})$ to the complexified K\"ahler moduli of $(Y,D)$ and prove that the special Lagrangian fibration on $(Y,D)$ is $T$-dual to the special Lagrangian fibration on $(\check{Y}, \check{D})$ previously constructed by the authors. We give some applications of these results, including to the study of automorphisms of del Pezzo surfaces fixing an anti-canonical divisor.
We study the deformed Hermitian-Yang-Mills equation on the blowup of complex projective space. Using symmetry, we express the equation as an ODE which can be solved using combinatorial methods if an algebraic stability condition is satisfied. This gives evidence towards a conjecture of the first author, T.C. Collins, and S.-T. Yau on general compact Kahler manifolds.
We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagrangian angle operator in the highest branch, we are able to construct continuous solutions of the associated Dirichlet problem.
We study the existence of special Lagrangian submanifolds of log Calabi-Yau manifolds equipped with the complete Ricci-flat K\"ahler metric constructed by Tian-Yau. We prove that if $X$ is a Tian-Yau manifold, and if the compact Calabi-Yau manifold at infinty admits a single special Lagrangian, then $X$ admits infinitely many disjoint special Lagrangians. In complex dimension $2$, we prove that if $Y$ is a del Pezzo surface, or a rational elliptic surface, and $D\in |-K_{Y}|$ is a smooth divisor with $D^2=d$, then $X= Y\backslash D$ admits a special Lagrangian torus fibration, as conjectured by Strominger-Yau-Zaslow and Auroux. In fact, we show that $X$ admits twin special Lagrangian fibrations, confirming a prediction of Leung-Yau. In the special case that $Y$ is a rational elliptic surface, or $Y= \mathbb{P}^2$ we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-K\"ahler rotation, $X$ can be compactified to the complement of a Kodaira type $I_{d}$ fiber appearing as a singular fiber in a rational elliptic surface $\check{\pi}: \check{Y}\rightarrow \mathbb{P}^1$.
We prove a convergence result for a family of Yang-Mills connections over an elliptic $K3$ surface $M$ as the fibers collapse. In particular, assume $M$ is projective, admits a section, and has singular fibers of Kodaira type $I_1$ and type $II$. Let $\Xi_{t_k}$ be a sequence of $SU(n)$ connections on a principal $SU(n)$ bundle over $M$, that are anti-self-dual with respect to a sequence of Ricci flat metrics collapsing the fibers of $M$. Given certain non-degeneracy assumptions on the spectral covers induced by $\bar\partial_{\Xi_{t_k}}$, we show that away from a finite number of fibers, the curvature $F_{\Xi_{t_k}}$ is locally bounded in $C^0$, the connections converge along a subsequence (and modulo unitary gauge change) in $L^p_1$ to a limiting $L^p_1$ connection $\Xi_0$, and the restriction of $\Xi_0$ to any fiber is $C^{1,\alpha}$ gauge equivalent to a flat connection with holomorphic structure determined by the sequence of spectral covers. Additionally, we relate the connections $\Xi_{t_k}$ to a converging family of special Lagrangian multi-sections in the mirror HyperK\"ahler structure, addressing a conjecture of Fukaya in this setting.
Let $X\rightarrow {\mathbb P}^1$ be an elliptically fibered $K3$ surface, admitting a sequence $\omega_{i}$ of Ricci-flat metrics collapsing the fibers. Let $V$ be a holomorphic $SU(n)$ bundle over $X$, stable with respect to $\omega_i$. Given the corresponding sequence $\Xi_i$ of Hermitian-Yang-Mills connections on $V$, we prove that, if $E$ is a generic fiber, the restricted sequence $\Xi_i|_{E}$ converges to a flat connection $A_0$. Furthermore, if the restriction $V|_E$ is of the form $\oplus_{j=1}^n\mathcal O_E(q_j-0)$ for $n$ distinct points $q_j\in E$, then these points uniquely determine $A_0$.
We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on $\mu$-polystable holomorphic bundles over $\mathbf{P}^{n-1}$.
We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical K\"ahler manifolds: If $\mathscr{E}$ is a reflexive sheaf over an ACyl K\"ahler manifold, which is asymptotic to a $\mu$-stable holomorphic vector bundle, then it admits an asymptotically translation-invariant protectively Hermitian Yang-Mills metrics (with curvature in $L^2_{\mathrm{loc}}$ across the singular set). Our proof combines the analytic continuity method of Uhlenbeck and Yau [UY86] with the geometric regularization scheme introduced by Bando and Siu [BS94].
Let $(X,\alpha)$ be a K\"ahler 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 \[ \sum_{i=1}^{n} \arctan(\lambda_i)= h(x) \] where $\lambda_i$ are the eigenvalues of $\omega$ with respect to $\alpha$. When $h(x)$ is a topological constant, this equation corresponds to the deformed Hermitian-Yang-Mills (dHYM) equation, and is related by Mirror Symmetry to the existence of special Lagrangian submanifolds of the mirror. We introduce a notion of subsolution for this equation, and prove a priori $C^{2,\beta}$ estimates when $|h|>(n-2)\frac{\pi}{2}$ and a subsolution exists. Using the method of continuity we show that the dHYM equation admits a smooth solution in the supercritical phase case, whenever a subsolution exists. Finally, we discover some stability-type cohomological obstructions to the existence of solutions to the dHYM equation and we conjecture that when these obstructions vanish the dHYM equation admits a solution. We confirm this conjecture for complex surfaces.
Let $L$ be a holomorphic line bundle over a compact K\"ahler manifold $X$. Motivated by mirror symmetry, we study the deformed Hermitian-Yang-Mills equation on $L$, which is the line bundle analogue of the special Lagrangian equation in the case that $X$ is Calabi-Yau. We show that this equation is the Euler-Lagrange equation for a positive functional, and that solutions are unique global minimizers. We provide a necessary and sufficient criterion for existence in the case that $X$ is a K\"ahler surface. For the higher dimensional cases, we introduce a line bundle version of the Lagrangian mean curvature flow, and prove convergence when $L$ is ample and $X$ has non-negative orthogonal bisectional curvature.
Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as the dimension reduction of the Hermitian-Yang-Mills equation for holomorphic vector bundles on K3 surfaces in the large complex structure limit. We define a notion of slope stability, and show that if the flat connection D has regular singularities, and the Riemannian metric g has finite volume then E admits a Poisson metric with asymptotics determined by the parabolic structure if and only if (E,D,P) is slope polystable.
We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X. We construct a natural barrier function along the flow, and introduce some techniques to study the blow-up of the curvature along the flow. Making some technical assumptions, we show how our techniques can be used to prove that the curvature of the evolved connection is uniformly bounded away from an analytic subvariety determined by the Harder-Narasimhan-Seshadri filtration of E. We also discuss how our assumptions are related to stability in some simple cases.
Let E_G be a principal G-bundle over a compact connected K\"ahler manifold, where G is a connected reductive complex linear algebraic group. We show that E_G is semistable if and only if it admits approximate Hermitian-Einstein structures.
Given a Sasaki manifold S, we prove the Sasaki-Ricci flow converges exponentially fast to a Sasaki-Einstein metric if one exists, provided the automorphism group of the transverse holomorphic structure is trivial.