Nonlinear analysis on manifolds
Org:
Siyuan Lu (McMaster) and
Jérôme Vétois (McGill)
[
PDF]
 SPYROS ALEXAKIS, University of Toronto
Singularity formation in Black hole interiors [PDF]

Starting from classical examples of singularity formation inside
black holes, I will recall the strong cosmic censorship conjecture of
Penrose regarding question. I will also review some further predictions
and known results on the generic behavior of the spacetime metric as it
terminates at a singularity; these results will be compared with the
complementary picture on initial, bigbang type singularities. The main
new result we will present is a recent proof of the perturbative stability
of the Schwarzschild singularity in vacuum, under polarized perturbations
of the initial data. The singularity that then forms is again of
spacelike character, and the solution displays
asymptoticallyvelocitytermdominated behavior upon approach to the
singularity. Joint with G. Fournodavlos.
 HUSSEIN CHEIKHALI, Université Libre de Bruxelles
The second best constant for the HardySobolev inequality on manifolds [PDF]

We consider the second best constant in the HardySobolev inequality on a Riemannian manifold. More precisely, we are interested with the existence of extremal functions for this inequality. This problem was tackled by DjadliDruet \cite{DD} for Sobolev inequalities. Here, we establish the corresponding result for the singular case. In addition, we perform a blowup analysis of solutions to HardySobolev equations of minimizing type. This yields informations on the value of the second best constant in the related Riemannian functional inequality.
\begin{thebibliography}{10}
\bibitem{DD}{Zindine Djadli and Olivier Druet, Extremal functions for optimal Sobolev inequalities on compact manifolds, Calc. Var. Partial Differential Equations {\bf 12} (2001), no. 1, 5884.}
\end{thebibliography}
 EDWARD CHERNYSH, McGill University
A global compactness theorem for critical pLaplace equations with weights [PDF]

In this talk, we investigate the compactness of PalaisSmale sequences for a class of critical $p$Laplace equations with weights. More precisely, we discuss a Struwetype decomposition result for PalaisSmale sequences, thereby extending a recent result of MercuriWillem (2010) to weighted equations. In sharp contrast to the model case of the unweighted critical $p$Laplace equation, all bubbling must occur at the origin. Furthermore, an adapted rescaling law is required to circumvent new difficulties introduced by the weights.
 BEOMJUN CHOI, University of Toronto / KIAS
Liouville theorem for surfaces translating by subaffinecritical powers of Gauss curvature [PDF]

We construct and classify the translating solutions to the flows by subaffinecritical powers of the Gauss curvature in $\mathbb{R}^3$. If $\alpha$ denotes the power, this corresponds to a Liouville theorem for degenerate MongeAmpere equations $\det D^2 u=(1+Du^2)^{2\frac{1}{2\alpha}}$ on $\mathbb{R}^2$ for $0<\alpha<1/4$. For the affinecritical case $\det D^2 u =1$, a classical result by J\"orgens, Calabi and Pogorelov shows the level curves of given solution are homothetic ellipses. In our case, the level curves converge asymptotically to a round circle or a curve with $k$fold symmetry for some $3\le k \le n_\alpha$. More precisely, these curves are closed shrinking curves to the $\frac{\alpha}{1\alpha}$curve shortening flow that were previously classified by Andrews in 2003. This is a joint work with K. Choi and S. Kim.
 SHUBHAM DWIVEDI, Humboldt University, Berlin
Deformation theory of nearly $\mathrm{G}_2$ manifolds [PDF]

We will discuss the deformation theory of nearly $\mathrm{G}_2$ manifolds. These are seven dimensional manifolds admitting real Killing spinors. After briefly discussing the preliminaries, we will show that the infinitesimal deformations of nearly $\mathrm{G}_2$ structures are obstructed in general. Explicitly, we will show that the infinitesimal deformations of the homogeneous
nearly $\mathrm{G}_2$ structure on the Aloff–Wallach space are all obstructed to second order. We will
also completely describe the cohomology of nearly $\mathrm{G}_2$ manifolds. This talk is based on a joint work with Ragini Singhal (University of Waterloo) (https://arxiv.org/abs/2007.02497).
 PENGFEI GUAN, McGill University
Locally constrained mean curvature type flows [PDF]

The talk concerns a class of mean curvature type flows with constraints. The first of such flow involving mean curvature was considered in a previous joint work with Junfang Li to provide a flow approach to the classical isoperimetric inequality. Later, general fully nonlinear constrained flows were introduced for optimal geometric inequalities involving quermassintegrals. These flows are associated with variational properties of corresponding geometric quantities. We will discuss some recent results and open regularity problems.
 ROBERT HASLHOFER, University of Toronto
Mean curvature flow through necksingularities [PDF]

In this talk, I will explain our recent work showing that mean curvature flow through necksingularities is unique. The key is a classification result for ancient asymptotically cylindrical flows that describes all possible blowup limits near a necksingularity. In particular, this confirms Ilmanen’s meanconvex neighborhood conjecture, and more precisely gives a canonical neighborhood theorem for necksingularities. Furthermore, assuming the multiplicityone conjecture, we conclude that for embedded twospheres mean curvature flow through singularities is wellposed. The twodimensional case is joint work with Choi and Hershkovits, and the higherdimensional case is joint with Choi, Hershkovits and White.
 HAN HONG, University of British Columbia
Stability and index estiamtes of capillary surfaces [PDF]

In this talk, we will discuss stability and index estimates for compact and noncompact capillary surfaces. A classical result in minimal surface theory says that a stable complete minimal surface in $\mathbb{R}^3$ must be a plane. We show that, under certain curvature assumptions, a strongly stable capillary surface in a 3manifold with boundary has only three possible topological configurations. In particular, we prove that a strongly stable capillary surface in a halfspace of $\mathbb{R}^3$ which is minimal or has the contact angle less than or equal to $\pi/2$ must be a halfplane. We also give index estimates for compact capillary surfaces in 3manifolds by using harmonic oneforms. This is joint work with Aiex and Saturnino.
 CHRISTOPHER KENNEDY, University of Toronto
A Bochner Formula on Path Space for the Ricci Flow [PDF]

Aaron Naber (Northwestern) and Robert Haslhofer (Toronto) have characterized solutions of the Einstein equation $\mathrm{Rc}(g) = \lambda g$ in terms of both sharp gradient estimates for Brownian motion and a Bochner formula on elliptic path space $PM$. They also successfully characterized solutions of the Ricci flow $\partial_{t}g = 2 \mathrm{Rc}(g)$ in terms of an infinitedimensional gradient estimate on parabolic path space $P\mathcal{M}$ of spacetime $\mathcal{M}=M \times [0,T]$.
\newline \In this talk, we shall generalize the classical Bochner formula for the heat flow on evolving manifolds $(M,g_{t})_{t \in [0,T]}$ to an infinitedimensional Bochner formula for martingales, thus proving the parabolic counterpart of recent results in the elliptic setting as well as characterizing solutions of the Ricci flow in terms of Bochner inequalities on parabolic path space. Timepermitting, we shall also discuss gradient and Hessian estimates for martingales on parabolic path space as well as a condensed proof of previous characterizations of the Ricci flow.
 YANGYANG LI, Princeton University
Generic Regularity of Minimal Hypersurfaces in Dimension 8 [PDF]

The wellknown Simons’ cone suggests that minimal hypersurfaces could be possibly singular in a Riemannian manifold with dimension greater than 7, unlike the low dimensional case. Nevertheless, it was conjectured that one could perturb away these singularities generically. In this talk, I will discuss how to perturb them away to obtain a smooth minimal hypersurface in an 8dimension closed manifold, by induction on the "capacity" of singular sets. This result generalizes the previous works by N. Smale and by ChodoshLiokumovichSpolaor to any 8dimensional closed manifold. This talk is based on joint work with Zhihan Wang.
 JIAWEI LIU, OttovonGuerickeUniversity Magdeburg
Ricci flow starting from an embedded closed convex surface in $\mathbb{R}^3$ [PDF]

We talk about the existence and uniqueness of Ricci flow that admits an embedded closed convex surface in $\mathbb{R}^3$ as metric initial condition. The main point is a family of smooth Ricci flows starting from smooth convex surfaces whose metrics converge uniformly to the metric of the initial surface in intrinsic sense. This is joint work with Jiuzhou Huang.
 HUSSEIN MESMAR, lorraine university IECL
Solution for HardySobolev equation in presence of isometrie [PDF]

Let $(M; g)$ be a smooth compact Riemannian manifold of dimension $ n \geq 4$, $G$ a closed subgroup
of the group of isometries $Isom_g(M)$ of $(M,g)$ and $k = \min_{x\in M} dim Gx$, where $Gx$ denotes the orbit of a point $x \in M$ under $G$. We fixe a point $x_0 \in M$ that $dim G{x_0}=k$ and $s \in (0; 2)$. We say that a function
$\phi : M \to \mathbb{R} $ is Ginvariant if $\phi (gx) = \phi (x)$ for any $x \in M$ and $g \in G$.
We investigate a suffcient
condition for the existence of a distributional continuous positive Ginvariant solution for
the HardySobolev equation
\[ \tag{E}
\Delta_g u + a u = \frac{u^{2^*(k,s)1}}{d_g(x,Gx_0)^s} + h u^{q1}
\]
where $\Delta_g: =  div_g ( \nabla )$ is the LaplaceBeltrami operator, ,$ a$, $h$ $ \in C^0 (M)$, $h \geq 0$,
$d_g$ is the Riemannian distance on $(M; g)$, $2^* (k,s) = \frac{2(nks)}{nk2}$ and $q \in (2, 2^*(k,s))$ with $2^* = 2^* (0,0)$. We prove that the existence of a Mountain Pass solution for the
above perturbative equation depends only on the perturbation. For that we need to prove first that for any $\epsilon > 0$, exist $A > 0$ and $B_\epsilon =B(\epsilon) \geq 0 $ so that for any $ u \in L^{2^*(k,s)}(M, d_g(x,Gx_0)^{s})$
\begin{align*}
\mathbb{\vert\vert} u \mathbb{\vert\vert}_{L^{2^*(k,s)}(M, d_g(x,Gx_0)^{s} ) }^2 \leq (A + \epsilon ) \mathbb{\vert\vert} \nabla u \mathbb{\vert\vert}_2^2 + B_\epsilon \mathbb{\vert\vert} u \mathbb{\vert\vert}_2^2
\end{align*}
 PENGZI MIAO, University of Miami
On interaction between scalar curvature and boundary mean curvature [PDF]

Scalar curvature and mean curvature are some of the most basic curvature quantities associated to a Riemannian manifold and its hypersurfaces, respectively. In a relativistic context, scalar curvature relates to matter distribution in a spacetime and mean curvature is used to compute the quasilocal mass of a finite body. In Riemannian geometry, existence and nonexistence of positive scalar curvature metrics is a fundamental question on closed manifolds. If the manifold is noncompact, important results on metrics with nonnegative scalar curvature include the Riemannian positive mass theorem and the Riemannian Penrose inequality. In this talk, we discuss how nonnegative scalar curvature in the interior of a compact manifold influences the mean curvature of its boundary hypersurface. Part of the talk is based on joint work with Siyuan Lu.
 ALEX MRAMOR, Johns Hopkins University
On the unknottedness of self shrinkers [PDF]

Self shrinkers are basic singularity models for the mean curvature flow. Much progress has been made in their study but outside some curvature convexity conditions and other special cases they are still not fully understood. In this talk I'll discuss some "unknottedness" results for self shrinkers in $\mathbb{R}^3$, which for instance imply that a self shrinking torus cannot be a tubular neighborhood of a nontrivial knot. The arguments discussed use the mean curvature flow and include some families of noncompact self shrinkers  closed self shrinkers were previously considered in a joint work with Shengwen Wang.
 KEATON NAFF, Columbia University
A local noncollapsing estimate for mean curvature flow [PDF]

We will discuss noncollapsing in mean curvature flow and prove a local version of the noncollapsing estimate. By combining our result with earlier work of X.J. Wang, it follows that certain ancient convex solutions that sweep out the entire space are noncollapsed. This is joint work with S. Brendle.
 JIEWON PARK, Caltech
The Laplace equation on noncompact Ricciflat manifolds [PDF]

We will discuss geometric applications of the Laplace equation on a complete Ricciflat manifold with Euclidean volume growth. We will focus on how to identify two arbitrarily far apart scales in the manifold in a natural way, exploiting the Łojasiewicz inequality of ColdingMinicozzi, in the case when a tangent cone at infinity has smooth cross section. We also prove a matrix Harnack inequality for the Green function when there is an additional condition on sectional curvature, which is an analogue of various matrix Harnack inequalities obtained by Hamilton and LiCao in different timedependent settings.
 SÉBASTIEN PICARD, UBC
Topological Transitions of CalabiYau Threefolds [PDF]

It was proposed in the works of Clemens, Reid and Friedman to connect CalabiYau threefolds with different topologies by a process known as a conifold transition. This operation may produce a nonKahler complex manifold with trivial canonical bundle. In this talk, we will discuss the propagation of differential geometric structures such as metrics with special holonomy and YangMills connections through conifold transitions. This is joint work with T. Collins and S.T. Yau.
 BRUNO PREMOSELLI, Université Libre de Bruxelles
Towers of bubbles for Yamabetype equations in dimensions larger than 7 [PDF]

In this talk we consider perturbations of Yamabetype equations on closed Riemannian manifolds. In dimensions larger than 7 and on locally conformally flat manifolds we construct blowingup solutions that behave like towers of bubbles (or bubbletrees) concentrating at a critical point of the mass function. Our result does not assume any symmetry on the underlying manifold.
We perform our construction by combining finitedimensional reduction methods with a linear blowup analysis. Our approach works both in the positive and signchanging case. As an application we prove the existence, on a generic bounded open set of $\mathbb{R}^n$, of blowingup solutions of the BrézisNirenberg equation that behave like towers of bubbles with alternating signs.
 FRÉDÉRIC ROBERT, Université de Lorraine
Blowingup solutions for secondorder critical elliptic equations: the impact of the scalar curvature [PDF]

Given a closed manifold $(M^n,g)$, $n\geq 3$, Olivier Druet proved that a necessary condition for the existence of energybounded blowingup solutions to perturbations of the equation
$$\Delta_gu+h_0u=u^{\frac{n+2}{n2}},\ u>0\hbox{ in }M$$
is that $h_0\in C^1(M)$ touches the Scalar curvature somewhere when $n\geq 4$ (the condition is different for $n=6$). In this paper, we prove that Druet's condition is also sufficient provided we add its natural differentiable version. For $n\geq 6$, our arguments are local. For the low dimensions $n\in\{4,5\}$, our proof requires the introduction of a suitable mass that is defined only where Druet's condition holds. This mass carries global information both on $h_0$ and $(M,g)$.
 XI SISI SHEN, Northwestern University
Estimates for metrics of constant Chern scalar curvature [PDF]

We discuss the existence problem of constant Chern scalar curvature metrics on a compact complex manifold. We prove a priori estimates for these metrics conditional on an upper bound on the entropy, extending a recent result by ChenCheng in the K\"ahler setting.
 VLADMIR SICCA, McGill University
A prescribed scalar and boundary mean curvature problem on compact manifolds with boundary [PDF]

In this talk I will present our recent result in the problem of finding a metric in a given conformal class with prescribed nonpositive scalar curvature and nonpositive boundary mean curvature, on a compact manifold with boundary. We established a necessary and sufficient condition in terms of a conformal invariant that measures the zero set of the target curvatures, which we call the relative Yamabe invariant of the set. (This is a joint work with Gantumur Tsogtgerel).
 FREID TONG, Columbia University
On the degenerations of asymptotically conical CalabiYau metrics [PDF]

The analytic study of complete noncompact Ricciflat Kahler metrics began with the work of TianYau in the 90s, who used PDE methods to produce many interesting examples of such metrics. In this talk, we will discuss the degenerations of noncompact Ricciflat Kahler metrics from an analytic point of view: by studying the limit of the corresponding complex MongeAmpere equations. In certain cases, we will see that the degenerate metric limit induces a complete singular Ricciflat Kahler metric on a quasiprojective variety and we will discuss the applications to constructions of complete Ricciflat Kahler metrics with singularities. This is joint work with T. Collins and B. Guo.
 JUNCHENG WEI, University of British Columbia
Sharp quantitative estimates for Struwe's decomposition [PDF]

Suppose $u\in D^{1,2} (\mathbb{R}^n)$. In a fundamental paper in 1984, Struwe proved that if $\Delta u+u^{\frac{2n}{n2}}_{H^{1}}:=\Gamma(u)\to 0$ then $\delta(u)\to 0$, where $\delta(u)$ denotes the $D^{1,2}(\mathbb{R}^n)$distance of $u$ from the manifold of sums of Talenti bubbles, i.e.
$$ \delta (u):=\inf_{\substack{(z_1,\cdots,z_\nu)\in \mathbb{R}^n\\ \lambda_1,\cdots,\lambda_\nu>0}}\left\\nabla u\nabla\left(\sum_{i=1}^{\nu} U\left[z_{i}, \lambda_{i}\right]\right)\right\_{L^{2}}.
$$
In 2019, Figalli and Glaudo obtained the first quantitative version of Struwe's decomposition in lower dimensions, namely $\delta(u)\lesssim \Gamma(u)$ when $3\leq n\leq 5$.
In this talk, I will present the following quantitative estimates of Struwe's decomposition in higher dimensions:
\[\delta (u)\leq C\begin{cases} \Gamma(u)\left\log \Gamma(u)\right^{\frac{1}{2}}\quad&\text{if }n=6,\ \Gamma(u)^{\frac{n+2}{2(n2)}}\quad&\text{if }n\geq 7.\end{cases}\]
Furthermore, we show that this inequality is sharp. (Joint work with B. Deng and L. Sun.)
 FENGRUI YANG, McGill University
Prescribed curvature measure problem in hyperbolic space [PDF]

The problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this talk, I will talk about prescribed curvature measure problem in hyperbolic space.
We establish the existence and regularity of solutions to the problem. The key is the $C^2$ regularity estimates for solutions to the corresponding fully nonlinear PDE in the hyperbolic space.
 SIYI ZHANG, University of Notre Dame
Conformally invariant rigidity theorems on fourmanifolds with boundary [PDF]

We introduce conformal and smooth invariants on oriented, compact fourmanifolds with boundary and show that "positivity" conditions on these invariants will impose topological restrictions on underlying manifolds with boundary. We also establish conformally invariant rigidity theorems for Bachflat fourmanifolds with boundary under the assumptions on these invariants. It is noteworthy to point out that we rule out some examples arising from the study of closed manifolds in the setting of manifolds with umbilic boundary.
 XIANGWEN ZHANG, UC Irvine
A geometric flow for Type IIA superstrings [PDF]

The equations of flux compactifications of Type IIA superstrings were written down by Tomasiello and TsengYau. To study these equations, we introduce a natural geometric flow on symplectic CalabiYau 6manifolds. We prove the wellposedness of this flow and establish the Shitype estimates which provides a criterion for the long time existence. As an application, we make use of our flow to find optimal almost complex structures on certain homogeneous symplectic halfflat manifolds. This is based on joint work with Fei, Phong and Picard.
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