Perspective in Geometric Analysis
Shing-Tung Yau
Harvard University

Tohoku University
July 12, 2016

In this talk, I shall discuss some directions in the subject ofgeometric analysis. I have written several times about the subject
in the past years. Many of my talks can be found in my selected
works on survey papers (International Press of Boston, 2014).
In this talk, I shall focus on several problems that may beinteresting to students who are new to the subject.

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1. General relativity

There are many interesting problems in this classical subject, which
we may say to be geometry of Lorentzian manifolds. However, weshould always remember that this is also a branch of physics. It
has to be compatible with the phenomena that we know about
gravity according to the basic principle of General relativity .
a. Within the subject of General relativity, a very fundamental
question is to find the right definition of quasi-local mass andquasi-local linear momentum.

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Quasi-local mass means that given a two dimensional spacelike
closed surface in spacetime, we want to associate a quantity which
satisfies several “axioms”, namely thati. It depends only on the metric of the surface, and the second
fundamental form of the surface.
ii. It is trivial if the surface is situated within the Minkowski
spacetime.
iii. It should be positive if the spacelike surface is convex in acertain sense and the spacetime satisfies the local energy
condition.

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iv. When a sequence of surfaces converges to the sphere atspatial infinity, the quasi-local mass should converge to theADM mass as was defined by Arnowit-Deser-Miser in thefollowing manner.
Z
1
MADM =lim(∂j gij − ∂j tr g )ν j
16π r →∞ S 2 (r )Here ν j is the outward unit normal of S 2 (r ).

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v. When a sequence of surfaces converges to the sphere at null
infinity, the quasi-local mass should converge to the Bondi
mass defined by Bondi et al. in (Proc. Roy. Soc. London A1962).
vi. When the surfaces converge to a point, it should reproduce
the matter density when the matter tensor is not zero,otherwise to the quantity related to Bel-Robinson tensor.
vii. When the space-time has global symmetric group given by
SU(2), the quasi-local mass of the orbit of SU(2) is equivalent
to the Komar mass defined by the method of Emmy Noether.

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The important question is whether such a definition is unique up
to equivalence or not? Mu-Tao Wang and I (CMP & PRL 2009)
succeeded to provide a construction of such a quasi-local mass.
Besides the definition given by Bartnik, this is the only known
definition of quasi-local mass that satisfy all of the above
properties.
I What is the relation between the Wang-Yau quasi-local mass

with Bartnik mass?
I The quasi-local mass that was defined Wang-Yau or Bartnik

are modeled after Minkowski spacetime. One can model thedefinition after other static spacetime. Can we find similarproperties?

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b. Isometric embeddings of surfaces into Minkowski spacetime. In
the works of Chen-Wang-Yau, we encountered the question ofoptimal isometric embedding into Minkowski spacetime. We knowthat the optimal isometric embedding is unique in many cases. It
will be good to find general conditions to establish uniqueness. It
is also interesting to know whether a general compact surface with
nonzero genus can be isometrically embedded into Minkowski
spacetime or not.

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c. Classification of complete spacelike hypersurfaces in
Schwarzschild spacetime with constant mean curvature. This wasdone by Andrejs Treibergs in (Invent. Math. 1982). It will be niceto find the detailed structure of its asymptotic structure even if weassume the mean curvature is equal to zero and the spacelike
hypersurface is asymptotic to a hyperplane that is boosted. Suchstructure can be very useful for many discussions of generalspacetime which is asymptotically flat.

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d. What is the fate of maximal spacelike hypersurfaces with
nontrivial second fundamental forms when when we evolve themaccording to the Einstein equation?

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2. Metric geometrya. Prove that when dimension is greater than 26, every compact
manifold with positive sectional curvature is diffeomorphic to ahomogeneous space.
b. Formulate and prove the Hamilton Inequality for Ricci flow
when the curvature is not necessarily positive.

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c. Consider the space of compact manifolds with positive scalar
curvature that is the conformal boundary of an Einstein manifold
which is asymptotic to hyperbolic space form. Can one do surgery
in codimension 3 for this class of manifolds, as was demonstrated
by Schoen-Yau within the class of metrics with positive scalar
curvature (cf. the paper of Gromov-Lawson). It is not hard todemonstrate that connected sum preserves this class of metrics.
Witten and I proved that the conformal boundary of such Einsteinmanifold must be connected. It seems to me that there is no newobstruction for such class of metrics than those from positive
scalar curvature alone.

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d. For a compact manifold with dimension greater than 4, do they
always an Einstein metric and in fact , would there be infinite
number of such metrics? If we impose extra structure on themetric such as Kählerian, we know that they are parametrized by
the space of Kählerian complex structures (due to the uniquenesstheorems developed by Calabi and Bando-Mabuchi).
We can weaken the requirement of Kählerness by requiring themetric to be a sum of Riemannian metrics that are locallyconformal to some Kähler metrics.

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3. Complex geometrya. On an n-dimensional Kähler manifold with nonvanishing
holomorphic n-form, there exists a Calabi-Yau metric andassociated with it, the concept of special Lagrangian submanifolds.
Prove that embedded special Lagrangian torus always exists. If
there is nontrivial deformation of the complex structure of theCalabi-Yau manifold, there is nontrivial family of special
Lagrangian torus that are mutually disjoint.
I Will this family fill up the Calabi-Yau manifold with someclosed exceptional set with codimension not less than two?
I What is the structures of this exceptional set?

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b. In the previous problem, if we look at the moduli space of thepairs (special Lagrangian torus vs. a flat complex line bundle overthis special Lagrangian torus).
In the paper of Strominger-Yau-Zaslow, this moduli space has acomplex Kähler structure and that after some “quantumcorrection”, it becomes a Calabi-Yau manifold mirror to theoriginal Calabi-Yau manifold. How to justify these claims?
If the special Lagrangian is nonsingular, the moduli space near it issmooth. The major question is to prove that we can complete thispart of moduli space to be a complex variety.

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c. Special Langragian submanifolds are supposed to be mirror tocertain holomorphic bundles defined in the mirror manifold. (Thebundle may have singularity. The rank of the bundle is theintersection number of the special Lagrangian submanifolds withthe special Lagrangian torus.)
Leung-Yau-Zaslow translated the equation of special Lagrangian toequation on the mirror bundle. The existence theory for thisequation was studied by Adam Jacob, Tristan Collins and me. It is
not completely understood. It is important to relate such existence
and the corresponding stability condition to the bridgeland stability.

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d. In studying eigenvalue of Laplacian on a projective manifold,Peter Li and I came up with a balanced condition for its projective
embedding. By changing the measure to be the induced measureof the ambient projective space, my former student Huazhang Luo
found that existence of such balanced condition implies Chow
stability of manifolds.
An important question is to find an effective way to projectively
move an embedded algebraic manifold to a balanced position. For
algebraic manifolds with ample canonical line bundle K , we knowthat by taking a power of K , the normalized induced metric atbalanced position will converge to the Kähler-Einstein metric when
the power of K tends to infinity. Can we compute theKähler-Einstein metric effectively by this method?

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4. Eigenvalues of elliptic operator and minimal surface

Both of these topics have long history. There are several
approaches to study eigenvalues of Laplacian which went back tomathematicians in 19th century through variational principle.
Important contributors include Kelvin, Rellich, Pólya, Szegö andothers.
Then a breakthrough happened in 1911 when Weyl solved afamous problem of Lorentz on asymptotic behavior of eigenvaluesof Laplacian. (Hilbert had thought that he would not see a proof
in his life time.) The argument used Tauberian theorem which waspopular in analytic number theory.

18 / 35This line of research developed into later works based on heat andwave equation methods. For manifolds whose universal cover issymmetric space, all eigenvalues and eigenfunctions are computed.
In this case, the method of Selberg is powerful and gives a lot of
information about eigenvalues related to the discrete groups acting
on symmetric space.

19 / 35Pólya conjecturea. Consider the Dirichlet problem for a bounded domain Ω in
Euclidean space Rn , Pólya (1961) conjectured that the kth
eigenvalue λk has a lower bound which is the same as the Weyl
asymtotics of λk , i.e.,λk ≥4π 2 k 2/n(Cn vol(Ω))2/n

Here Cn is the volume of the unit ball in Rn .
Peter Li and I (CMP 1983) proved Pólya conjecture in the average.
But the conjecture is still open.

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b. For a compact surface M, Peter Li and I (Invent. Math. 1982)
proved that the first eigenvalue times the area of the surface isbounded above by twice the conformal area.
I Compute the conformal area as a function defined on themoduli space of the Riemann surface.
I Is the Li-Yau upper bound optimal? For a fixed conformalstructure, what is the optimal metric to optimize the Li-Yauinequality?
I For some special surface such as those defined by arithmeticgroup, the lower bound also holds qualitatively as wasdemonstrated by Selberg’s famous 3/16 lower estimate. Whatis the conformal area of such surfaces that are defined byarithmetic group quotient?

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If we fix the conformal structure, and maximize the product of first
eigenvalue with area for metrics within that conformal class, whatis the result compared with the conformal area defined by Li-Yau?
If one does not fix the conformal structure, the problem of
extremal metric was studied by Nadirashvili (GAFA 1996) and inrecent works of Schoen and Fraser.
For higher dimensional Kähler manifold, we can ask similar
question of optimizing the first eigenvalue among Kähler metrics
within the same Kähler class. Similar questions can be asked forhigher eigenvalues.

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Suppose we fix the Kähler class as a cohomology class, and weoptimize eigenvalues of the Laplacian, we obtain infinite number of
numbers that are invariants of the complex structure and theKähler class. Can these numbers determine the complex structure
and the Kähler class?
Generalize my work with Peter Li and Bourguignon to Sasakian
manifolds.
I Can we map a Sasakian manifold to odd dimensional spherepreserving transverse holomorphic structure?
I Can we generalize the above discussions to balanced metrics?
I Is any of these discussions still valid for Laplacian acting on

differential forms?

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c. Can we estimate lower bound of the first eigenvalue of
Laplacian acting on differential forms in terms of isometricinequalities for subvarieties that are homologous to zero. Hencethe volume of the subvariety compared with the volume of the area
minimizing subvarieties that it bounds.

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d. In previous problem, we propose a sequence of numbers within afixed a Kähler class and complex structure. It may be called TypeB eigenvalues of the Laplacian. Is there a mirror picture of thesenumbers according to mirror principle for Calabi-Yau manifolds? Inthat case, we are looking for eigenvalues related to the symplectic
structure.
I Is there natural operator associate to the symplectic structure?
I If we fixed a symplectic form, then for each almost complex

structure that is compatible with the symplectic form, wehave Laplacian which gives rise to eigenvalues. Can we findsome inequality associated to these eigenvalues?

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5. Submanifolds
a. For which Calabi-Yau manifold and which integral homology
class of (k, k)-type, can it be represented as positive sum ofholomorphic cycles? This assertion refines the famous Hodge
conjecture and is not understood even when the manifold has
dimension three. Strictly speaking, there is counterexample due toWolfson for some K3 surfaces.
The general idea was to minimize area and attempt to prove theminimial variety is in fact a sum of holomorphic cycles. This idea
was effective when the cycle is 2-dimensional and has genus zeroas was demonstrated by Siu and Yau in the proof of the Frenkel
conjecture, by studying the second variational formula. The idea ofSiu-Yau was used later by Micallef-Moore and Brendle to treat
surfaces of high codimension in Riemannian manifolds.

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b. Classify embedded minimial surfaces in the three sphere.
Several mathematicians including Lawson, Meeks, Rosenberg,
Schoen et al had basically accomplished the classifications ofcompleted embedded surfaces in Euclidean space by reducing mostproblems to those minimal surfaces with finite total curvature
where algebraic geometry method can be brought in. Perhaps we
can hope to classify such surfaces in 3-sphere by algebro-geometricmeans?

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c. Consider closed minimal hypersurfaces M ⊂ S n+1 with constantscalar curvature κ. Chern conjectured that for each n the set of all
possible values for κ is discrete. It has not been completely solved
despite much progress was made.
One way of generalizing the Chern conjecture as originallyproposed is the following: Assume furthermore that some scalar
invariant of the second fundamental form h, or its covariant
derivatives, is constant on M (|h|2 in the case of the Chern
conjecture, other examples would be det h and |∇k h|2 ). Then one
can ask what values the constant can assume, in particularwhether the set of these values is discrete.
It is also possible to generalize questions of this type tosubmanifolds of higher codimension.

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One can of course ask similar question for compact minimal
submanifolds of constant scalar curvature in other compact
Riemannian homogeneous spaces. What kind of values are their
possible scalar curvatures? One can also ask the same question forsubmanifolds where some symmetric polynomials of the second
fundamental form is constant.

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d. Given a 3-dimensional Riemannian manifold M, can we find agraph of it in M × R so that this graph can be conformallyembedded into R4 . This is a question that arises from mydiscussion with Mu-Tao Wang and Po-Ning Chen on quasi-localmass.

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6. Special structure on manifolds

a. Understand which compact connected Lie group can beholonomy group of some Riemannina manifold. The discrete partis more complicated and is related to fundamental group of themanifold. It will be nice to find a systematic way to understandthis problem.
The question is reasonably understood for manifolds with constantpositive or zero curvature. They are covered by spheres or torus. In
the first case, it is called Vincent problem and solved by J. A.
Wolf. The second case was started by Bieberbach, but notcompletely understood.

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The negative curvature case is much more difficult. They aretorsion-free discrete cocompact subgroups of SO(n, 1). A great
progress was accomplished by Thurston’s geometrization theorem
which says that every compact 3-dimensional irreducible asphericaland atoroidal 3-manifold must admit a metric with constantcurvature. Which finitely presented torsion free atoroidal group canappear as fundamental group of compact manifold with negative
curvature? Of course, we know there are various kind ofhyperbolicity that can attach to it as was initiated by the famous
paper of Milnor.
The same question can be asked for other compact locallysymmetric space, positive curvature or not. Can we decide whichsmooth manifold admits such a structure?

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b. Berger has classified all manifolds with compact connected Liegroup as holonomy group. They can also be realized by examples.
However those with holonomy group Spin(7) and G2 are notunderstood as their moduli spaces have not been understood welland we cannot tell which 7 or 8-dimensional manifolds can support
such structures.
It is an important question to reduce the problem of existence ofsuch structures based on simple criterion. Donaldson and Thomas
introduced special bundles over such manifolds. It should be
important to understand their moduli space also.

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c. Classify those Lorenzian manifolds whose holonomic groups areproper subgroups of SO(n, 1). What kind of group can occur asholonomy group?
d. Let M be a compact complex manifold that admits a balanced
metric such that the tangent bundle is slope stable with respect tothe balanced metric. Supposed the Chern number inequality
becomes equality, what can we say about such balanced manifolds?

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Thank you!

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