A Brief Survey of Some Asympotics in the Study of Minimal ...scgas/scgas-2012/Talks/Simon.pdf · divXD R M HXwith sup MjHj

A Brief Survey of Some Asympotics in the Study of Minimal Submanifolds

Leon Simon

Stanford University

“A Brief Survey of Asympotics in the Study of Minimal Submanifolds” L. Simon

1.1

1 Preliminaries

MATHEMATICS DEPARTMENTSTANFORD UNIVERSITY

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� Preliminaries� Examples� Tangent Cones� Monotonicity� Approaches: “A”& “B”� Prototypical example� Caution� Prototypical ex. (cont.)� Realistic Considerations� Liapunov-Schmidt� Łojasiewicz in dim1� Direct Method App I� Direct Method App II� Blowup Methods� General Decay Lem.

Suppose N is some complete Riemannian manifold and M aclosed rectifiable subset of N which is “minimal” (i.e. stationarywith respect to the area functional)—that is d

dtjtD0Hn.'t.M// D 0

for any family of maps 't W N ! N with the properties that 'tvaries smoothly with t 2 .�1; 1/, '0 D identity and 9 compact Kwith 't jN nK D identity on N nK 8t 2 .�1; 1/.If N D Rp we can take 't.x/ D x C tX.x/ where X W Rp ! Rpis a smooth function with compact support, and in this case theabove stationarity gives the first variation formulaR

MdivM X D 0;

where divM means the tangential divergence of X relative to M ;thus divM X jx D

PniD1 �i �D�iX jx where �1; : : : ; �n is an orthonor-

mal basis for TxM .(In case the ambient manifold is a N rather than Rp we get anidentity of the form

RM

divX DRMH �X with supM jH j <1.)

We could also allow M to have multiplicity, but, except whereotherwise explicitly indicated, here we will typically work withmultiplicity 1 submanifolds.

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1.2

1 Preliminaries


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dtjtD0Hn.'t.M// D 0


MdivM X D 0;




RM



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1.3

1 Preliminaries


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dtjtD0Hn.'t.M// D 0


MdivM X D 0;




RM



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2.1

2 Preliminarides (Cont.)—Examples


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Of course there are many examples of such minimal submani-folds. One class which includes many singular examples is thegiven by the set of complex analytic varieties in Cp � R2p—for example let M0 D f �1.0/ where f W Cp ! C is holo-morphic, and let M � R2p be the corresponding real analyticvariety. Then (using a calibration argument) one can checkthat M is area minimizing, so that with n D 2p � 2 we haveHn.'t.M \ BR// � Hn.M \ BR/, hence minimal. In this case thesingular set “stratifies” into a locally finite union of lower (even)dimensional submanifolds.There is also a rich class of smooth (i.e. singular set D ;)minimal submanifolds obtained by PDE methods: For exam-ple quasilinear elliptic PDE theory tells us that we can solvethe minimal surface equation (i.e. the equation �u�

Pni;jD1.1C

jDuj2/�1DiuDjuDiDju D 0) on a closed ball BR with arbitraryprescribed continuous boundary data ', and the solution u iscontinuous on BR and real analytic in MBR D BR n @BR.

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2.2



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Of course there are many examples of such minimal submani-folds. One class which includes many singular examples is thegiven by the set of complex analytic varieties in Cp � R2p—for example let M0 D f �1.0/ where f W Cp ! C is holo-morphic, and let M � R2p be the corresponding real analyticvariety. Then (using a calibration argument) one can checkthat M is area minimizing, so that with n D 2p � 2 we haveHn.'t.M \ BR// � Hn.M \ BR/, hence minimal. In this case thesingular set “stratifies” into a locally finite union of lower (even)dimensional submanifolds.There is also a rich class of smooth (i.e. singular set D ;)minimal submanifolds obtained by PDE methods: For exam-ple quasilinear elliptic PDE theory tells us that we can solvethe minimal surface equation (i.e. the equation �u�

Pni;jD1.1C

jDuj2/�1DiuDjuDiDju D 0) on a closed ball BR with arbitraryprescribed continuous boundary data ', and the solution u iscontinuous on BR and real analytic in MBR D BR n @BR.

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There are many open questions about the structure of the singu-lar set of singular minimal submanifolds, including some verybasic ones: e.g. not known if there can be a minimal M with asequence of xn of isolated singular points such that xn! p 2M .Such behavior does not occur in real analytic varieties, so in par-ticular the singular examples discussed on the previous slidecannot not exhibit behavior of this type.

Indeed all of the known examples of singular minimal submani-folds have singular sets which “stratify” into locally finite unionsof submanifolds of lower dimension, and it is an open questionwhether or not such behaviour is generic in some reasonablesense. For instance: can one have an example of a minimal sub-manifold whose singular set is a (closed) fractional dimensionalsubset of a straight line or a C 1 curve? Or even simpler ques-tions: can the singular set consist of the union of two disjointclosed subintervals of a straight line in some Euclidean space.

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There are many open questions about the structure of the singu-lar set of singular minimal submanifolds, including some verybasic ones: e.g. not known if there can be a minimal M with asequence of xn of isolated singular points such that xn! p 2M .Such behavior does not occur in real analytic varieties, so in par-ticular the singular examples discussed on the previous slidecannot not exhibit behavior of this type.

Indeed all of the known examples of singular minimal submani-folds have singular sets which “stratify” into locally finite unionsof submanifolds of lower dimension, and it is an open questionwhether or not such behaviour is generic in some reasonablesense. For instance: can one have an example of a minimal sub-manifold whose singular set is a (closed) fractional dimensionalsubset of a straight line or a C 1 curve? Or even simpler ques-tions: can the singular set consist of the union of two disjointclosed subintervals of a straight line in some Euclidean space.

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3.1

3 Preliminaries (Cont.): Tangent Cones


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Aside from its intrinsic interest, an understanding of asymptoticbehavior is key to understanding the structure of the singularset—General Principle: Sufficiently precise information aboutasymptotic behavior on approach to singularities) good infor-mation about the structure of the singular set.M always looks “asymptotically conic” at all sufficiently smallscales near a singular point, as we discuss below.

The asymptotically conic nature of minimal submanifolds nearsingular points is of key importance, but its direct usefulness inanalyzing singularities is severely limited by the possible non-uniqueness of the limiting cones; i.e. examples that behave anal-ogous to the logarithmic spiral .r/ D rei

pj log r j; 0 < r < 1 (in

terms polar coordinates .r; �/, � Dpj log r j, which spirals into

the origin, and is close to a ray at each sufficiently small scale).Check: r; � 2 .0; 1/ ) j .r/ � .� r/j � j

pj log r j C j log � j �p

j log r jj D j log � j=.jpj log r j C

pj log r j C j log � jj/ �

j log � j=jpj log r j ! 0 as r # 0 regardless of how small � is.

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the origin, and is close to a ray at each sufficiently small scale).Check: r; � 2 .0; 1/ ) j .r/ � .� r/j � j

pj log r j C j log � j �p

j log r jj D j log � j=.jpj log r j C

pj log r j C j log � jj/ �

j log � j=jpj log r j ! 0 as r # 0 regardless of how small � is.

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the origin, and is close to a ray at each sufficiently small scale).

Check: r; � 2 .0; 1/ ) j .r/ � .�r/j � jpj log r j C j log �j �p

j log r jj D j log �j=.jpj log r j C

pj log r j C j log �jj/ �

j log �j=jpj log r j ! 0 as r # 0 regardless of how small � is.

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4.1

4 Preliminaries (Cont.): Monotonicity


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A key tool in the analysis of asymptotics on approach to a sin-gular point is the “monotonicity identity” (obtained by taking aradial deformation X jx D '.jxj/x in the first variation formula).

��njM \B�.´/j � ��njM \B�.´/j D

RM\.B�.´/nB� .´//

j.x � ´/?j2

jx � ´jnC2.� 0/

In particular ‚.´/ D lim�#0 ��njM \ B�.´/j exists and

��njM \ B�.´/j �‚.´/ DRM\B�.´/

j.x � ´/?j2

jx � ´jnC2

In particularRM\.B�.´/nB� .´//

j.x�´/?j

jx�´jnC2< 1, which is far from ob-

vious without the help of the above formula, and in fact saysthat in some L1 sense .jx � ´j�1.x � ´//? ! 0, which stronglysuggests the asymptotically conic nature of M at ´. (However,to formally check that one has to work a little harder, using atechnical variant of the above identity.)

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4.2

4 Preliminaries (Cont.): Monotonicity


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A key tool in the analysis of asymptotics on approach to a sin-gular point is the “monotonicity identity” (obtained by taking aradial deformation X jx D '.jxj/x in the first variation formula).

��njM \B�.´/j � ��njM \B�.´/j D

RM\.B�.´/nB� .´//

j.x � ´/?j2

jx � ´jnC2.� 0/

In particular ‚.´/ D lim�#0 ��njM \ B�.´/j exists and

��njM \ B�.´/j �‚.´/ DRM\B�.´/

j.x � ´/?j2

jx � ´jnC2

In particularRM\.B�.´/nB� .´//

j.x�´/?j

jx�´jnC2< 1, which is far from ob-

vious without the help of the above formula, and in fact saysthat in some L1 sense .jx � ´j�1.x � ´//? ! 0, which stronglysuggests the asymptotically conic nature of M at ´. (However,to formally check that one has to work a little harder, using atechnical variant of the above identity.)

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5.1

5 Methods


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Here we consider two approaches to the analysis of asymptoticbehavior:A. “Direct Method” Using direct analytic inequalities on thenon-linear functional (in particular an infinite dimensional ver-sion of some inequalities for the gradient of a real analytic func-tion due to Łojasiewicz)B. “Blowup Method” Using compactness arguments to provegood approximation to solutions of the relevant non-linear prob-lem by using solutions of the corresponding linearized operator.

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6.1

6 Direct Method Prototype


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A good prototypical problem for Approach A is the analysis ofasympotics at t D1 for the ODE system

.�/ P� D �rf .�/CR

where f is a (real-valued) real analytic on Rn and where R is“lower order” in the sense that 9 fixed � 2 .0; 1/ with jRj � � j P�j.

Łojasiewicz proved, using stratification theory for real analyticvarieties, that if �0 is a critical point for f (i.e. rf .�0/ D 0) andf .�0/ D 0, then 9˛ 2 .0; 1

2� and C; � > 0 such that

(Ł) jrjf j˛.x/j2 � C; x 2 B�.�0/ n {x W f .x/ ¤ 0}

(Trivial example: If f .x/ D jxj2 then jrf 1=2j D jrjxjj � 18x.)

Claim: Using (Ł) it is easy to prove �.t/ has a limit �0 as t !1and that j�.t/ � �0j � Ct�ˇ for some ˇ > 0 (i.e. power decayasymptotically), provided we assume that � remains boundedon the entire interval Œ0;1/:

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6.2

6 Direct Method Prototype


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A good prototypical problem for Approach A is the analysis ofasympotics at t D1 for the ODE system

.�/ P� D �rf .�/CR

where f is a (real-valued) real analytic on Rn and where R is“lower order” in the sense that 9 fixed � 2 .0; 1/ with jRj � � j P�j.

Łojasiewicz proved, using stratification theory for real analyticvarieties, that if �0 is a critical point for f (i.e. rf .�0/ D 0) andf .�0/ D 0, then 9˛ 2 .0; 1

2� and C; � > 0 such that

(Ł) jrjf j˛.x/j2 � C; x 2 B�.�0/ n {x W f .x/ ¤ 0}

(Trivial example: If f .x/ D jxj2 then jrf 1=2j D jrjxjj � 18x.)

Claim: Using (Ł) it is easy to prove �.t/ has a limit �0 as t !1and that j�.t/ � �0j � Ct�ˇ for some ˇ > 0 (i.e. power decayasymptotically), provided we assume that � remains boundedon the entire interval Œ0;1/:

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7.1

7 Caution


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Notice particularly that such a result is false if f is merelysmooth rather than real analytic; in the smooth case there areexamples where the set of critical points are e.g. a circle and �.t/just spirals in to that circle asymptotically as t !1, so in thatcase the set of limit points of �.t/ is a circle:

Possible picture of curve of steepest descent ( P� D �rf .�/) for C1 function f

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8.1

8 Direct Method Prototype (Cont.)


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Checking the Claim: Observe that � ddtf .�.t// D �rf .�.t// � P� D

P�.t/ � . P� CR/ � .1 � �/j P�j2 so

.1 � �/R1

tj P�j2 � f .�.t//

d

dt.f ˛�1.�.t/// D �.1 � ˛/f ˛�2.�/rf .�/ � P�

D .1 � ˛/f ˛�2.�/rf .�/ � .rf .�/ �R/

� Cf ˛�2.�/jrf .�/j2 D C jrf ˛=2.�/j2 � C

by (Ł) if �.t/ 2 B�.�0/. Integration) power decayR1

tj P�j2 � Cf .�.t// � Ct�1�ˇ and hence

R1

tj P�j � Ct�ˇ=2

8 t such that �.t/ 2 B�.�0/. Integration over intervals Œt1; t2�)

j�.t2/ � �.t1/j D jR t2t1P�j �

R t2t1j P�j � Ct

�ˇ=21

if �jŒt1; t2� � B�.�0/, so if �0 is a limit point of some sequence �.tk/(for some sequence tk !1) then �0 D limt!1 �.t/ and j�.t/��0j �Ct�ˇ=2. (i.e. power decay to the asymptotic limit).

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P�.t/ � . P� CR/ � .1 � �/j P�j2 so

.1 � �/R1

tj P�j2 � f .�.t//

d

dt.f ˛�1.�.t/// D �.1 � ˛/f ˛�2.�/rf .�/ � P�

D .1 � ˛/f ˛�2.�/rf .�/ � .rf .�/ �R/

� Cf ˛�2.�/jrf .�/j2 D C jrf ˛=2.�/j2 � C



R1



j�.t2/ � �.t1/j D jR t2t1P�j �


�ˇ=21


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P�.t/ � . P� CR/ � .1 � �/j P�j2 so

.1 � �/R1

tj P�j2 � f .�.t//

d

dt.f ˛�1.�.t/// D �.1 � ˛/f ˛�2.�/rf .�/ � P�

D .1 � ˛/f ˛�2.�/rf .�/ � .rf .�/ �R/

� Cf ˛�2.�/jrf .�/j2 D C jrf ˛=2.�/j2 � C



R1



j�.t2/ � �.t1/j D jR t2t1P�j �


�ˇ=21


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9.1

9 Applying the Direct Method to Minimal Submanifolds


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Of course the prototypical example above is not completely con-vincing for 2 reasons:(1) Minimal surfaces, at least away from their singular points,are locally represented as graphs of smooth functions—i.e. aninfinite dimensional setting, so to generalize from the prototyp-ical case to the actual case we need, at least, a Łojasiewicz in-equality in an appropriate infinite dimensional setting.(2) The relevant equations in the infinite dimensional setting (aswe’ll see below) are more analogous to the second order systemR� � P� D rf .�/CR rather than �P� D rf .�/CR.It turns out that in fact (2) is not a very serious worry, since acareful analysis shows that the essential difficulties center onthe “slowly varying” case when j R�j << j P�j, which makes it possi-ble to write R�� P� D rf .�/CR in the form �P� D rf .�/C zR wherej zRj < z� with z� a fixed constant in .0; 1/. Indeed using a trickinvolving a variant of the argument in the prototypical case dis-cussed above (via the monotonicity formula) one can avoid com-pletely any necessity to even address point (2). So (1) is the onlyproblem we need concern ourselves with.

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9.2

9 Applying the Direct Method to Minimal Submanifolds


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Of course the prototypical example above is not completely con-vincing for 2 reasons:(1) Minimal surfaces, at least away from their singular points,are locally represented as graphs of smooth functions—i.e. aninfinite dimensional setting, so to generalize from the prototyp-ical case to the actual case we need, at least, a Łojasiewicz in-equality in an appropriate infinite dimensional setting.(2) The relevant equations in the infinite dimensional setting (aswe’ll see below) are more analogous to the second order systemR� � P� D rf .�/CR rather than �P� D rf .�/CR.It turns out that in fact (2) is not a very serious worry, since acareful analysis shows that the essential difficulties center onthe “slowly varying” case when j R�j << j P�j, which makes it possi-ble to write R�� P� D rf .�/CR in the form �P� D rf .�/C zR wherej zRj < z� with z� a fixed constant in .0; 1/. Indeed using a trickinvolving a variant of the argument in the prototypical case dis-cussed above (via the monotonicity formula) one can avoid com-pletely any necessity to even address point (2). So (1) is the onlyproblem we need concern ourselves with.

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10.1

10 Liapunov-Schmidt Reduction


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In order to extend the Łojasiewicz inequality to the appropri-ate infinite dimensional setting, we need an appropriate versionof the Liapunov-Schmidt procedure (a procedure which worksvia the inverse function theorem and enables suitable infinitedimensional problems to be reduced to corresponding finite di-mensional ones).We consider a functional

F.u/ DR†F.!; u;ru/

defined on smooth sections u D u.!/ .! 2 †/ of the normal bun-dle over †, with † a smooth compact Riemannian manifold ofdimension m � 1, and F is assumed to be smooth in its de-pendence on !; u;ru. (We’ll need to assume that F is in factreal analytic in its dependence on u;ru in the discussion of theŁojasiewicz inequality for F below.)M.u/ (or “�rF.u/”) denotes the (second order) Euler-Lagrangeoperator (or “first variation operator”) of F characterized by

hM.u/; �iL2 DddsjsD0F.uC s�/:

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10.2



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10.4



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L.u/ D ddsjsD0M.su/

is the linearization of M at 0, assumed to be elliptic, so that L DkernelL is finite dimensional, with orthonormal basis '1; : : : ; 'q(relative to L2.†/), hence

PL.u/ DPqjD1�j'j .�j D hu; 'j iL2.†//

is the orthogonal projection onto L.Then

N .u/ D PL.u/CM.u/

has trivial kernel, so N has a smooth (or real-analytic if F isreal analytic in its dependence on u;ru) inverse ‰ defined in anhd. of 0. Then ‰.N .u// � u and hence

M.u/ D 0 ” u D ‰.PL.u// ” u 2M

in a suitable nhd. of 0, where M is the smooth (or real analytic)manifold of dimension q.

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Thus we have the following elegant and very useful picture:

Liapunov-Schmidt Reduction

The blue curves are contained in the manifold M , and repre-sent the set of all possible solutions of the non-linear equationM.u/ D 0 for small u. Some 1-variable calculus computationsalong line segments in L and elliptic estimates (C 2;˛ and W 2;2

estimates in fact) shows these further characterized by:

M.u/ D 0 iff u D ‰.P�j'j /withrf .�/ D 0; f .�/ D F.‰.

PnjD1�j'j //

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and in addition

C�1kM.u/kL2.†/ � jrf .�/j � CkM.u/kL2.†/

jF.u/ � f .�/j � CkM.u/k2L2.†/

;

for all sufficiently small u, where we continue to use the notationthat � D .�1; : : : ; �q/ 2 Rq with �j D hu; 'j iL2.†/.

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11.1

11 Infinite Dimensional Łojasiewicz Inequality


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As mentioned above, if F (the integrand in the functional F) hasreal analytic dependence on u;ru then f .�/ is a real analyticfunction of �, so we can use the Łojasiewicz inequality:

jf .�/ � f .0/j1�˛ � C jrf .�/j; j�j < �;

for suitable ˛ 2 .0; 12� and C; � > 0, and then the last two in-

equalities on the previous slide imply

jF.u/ � F.0/j1�˛ � CkM.u/kL2.†/

for u in a small enough (C 3) neighborhood of 0 (with the same ˛but slightly larger C ). This is the infinite dimensional versionof the Łojasiewicz inequality.In the applications to minimal submanifolds discussed below,we use the special case† D C\SN , where C is an n-dimensionalminimal cone in RNC1 with vertex at 0 and no other singulari-ties, and in this case we take F to be the area functional on †.Thus we are in the above setting with m D n � 1 and

F.u/ D Hn�1.graphu/

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11.2

11 Infinite Dimensional Łojasiewicz Inequality


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As mentioned above, if F (the integrand in the functional F) hasreal analytic dependence on u;ru then f .�/ is a real analyticfunction of �, so we can use the Łojasiewicz inequality:

jf .�/ � f .0/j1�˛ � C jrf .�/j; j�j < �;

for suitable ˛ 2 .0; 12� and C; � > 0, and then the last two in-

equalities on the previous slide imply

jF.u/ � F.0/j1�˛ � CkM.u/kL2.†/

for u in a small enough (C 3) neighborhood of 0 (with the same ˛but slightly larger C ). This is the infinite dimensional versionof the Łojasiewicz inequality.In the applications to minimal submanifolds discussed below,we use the special case† D C\SN , where C is an n-dimensionalminimal cone in RNC1 with vertex at 0 and no other singulari-ties, and in this case we take F to be the area functional on †.Thus we are in the above setting with m D n � 1 and

F.u/ D Hn�1.graphu/

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12.1

12 Uniqueness of Tangent Cone Theorem via Direct Method


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Theorem. If M is an n-dimensional minimal submanifoldof Rp with 0 2 singM , and if C is one of the tangent conesof M at 0 (so that ��1k M converges, in the appropriate weakmeasure sense, to C for some sequence �k # 0, then C is theunique tangent cone, the singularity of M at 0 is isolated andM is asymptotic to C at rate C jxj�ˇ.

To prove this pick a starting value � > 0 so that ��1M is mea-sure theoretically close (within some arbitrary " > 0) to C inan annular region � < r D jxj < 1. Now using the Allardregularity theorem ��1M must be close to C in the C 2 sensein the smaller annular region 2� < r < 1

2in the sense that

M \ {x W 2� < jxj < 12} D graphu with u a C 2 section of small

norm of the normal bundle over C\ {x W 2� < jxj < 12}, and with

u satisfying the Minimal Surface Equation MC.u/ D 0 (Euler-Lagrange equation for the area functional over C).

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Theorem. If M is an n-dimensional minimal submanifoldof Rp with 0 2 singM , and if C is one of the tangent conesof M at 0 (so that ��1k M converges, in the appropriate weakmeasure sense, to C for some sequence �k # 0, then C is theunique tangent cone, the singularity of M at 0 is isolated andM is asymptotic to C at rate C jxj�ˇ.

To prove this pick a starting value � > 0 so that ��1M is mea-sure theoretically close (within some arbitrary " > 0) to C inan annular region � < r D jxj < 1. Now using the Allardregularity theorem ��1M must be close to C in the C 2 sensein the smaller annular region 2� < r < 1

2in the sense that

M \ {x W 2� < jxj < 12} D graphu with u a C 2 section of small

norm of the normal bundle over C\ {x W 2� < jxj < 12}, and with

u satisfying the Minimal Surface Equation MC.u/ D 0 (Euler-Lagrange equation for the area functional over C).

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In terms of coordinates t D � log r and ! D x=jxj 2 † D C\Sp�1

one can check that MC.u/ D 0 can be written in the form

Ru � n Pu DM†.u/CR

where M† is the Euler-Lagrange operator for the areafunctional on † (so the infinite dimensional version of theŁojasiewicz inequality can be applied), and jRj � C".j Ruj C j Puj/.The initial proof of uniqueness of C did follow this approach,but that has been superseded by a more direct method using aslight variant of the monotonicity formula and the infinite di-mensional version of the Łojasiewicz inequality. (Still very rem-iniscent of the argument we used in the prototypical case.)The variant of the monotonicity formula mentioned above isthat (by extra step in the previous monotonicity discussion)R

M\B�.´/j.x�´/?j2

jx�´jnC2� C.n/.Hn�1.M \ @B�/ �Hn�1.†//

and the right side here is exactly C.F.u�/�F.0//, where u�.!/ D��1u.�!/ and F is the area functional of †, and it is possible tocomplete the argument essentially as in the prototypical case.

� � ! IO 12.3 (SLIDE 12/17)


12.4



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In terms of coordinates t D � log r and ! D x=jxj 2 † D C\Sp�1

one can check that MC.u/ D 0 can be written in the form

Ru � n Pu DM†.u/CR

where M† is the Euler-Lagrange operator for the areafunctional on † (so the infinite dimensional version of theŁojasiewicz inequality can be applied), and jRj � C".j Ruj C j Puj/.The initial proof of uniqueness of C did follow this approach,but that has been superseded by a more direct method using aslight variant of the monotonicity formula and the infinite di-mensional version of the Łojasiewicz inequality. (Still very rem-iniscent of the argument we used in the prototypical case.)The variant of the monotonicity formula mentioned above isthat (by extra step in the previous monotonicity discussion)R

M\B�.´/j.x�´/?j2

jx�´jnC2� C.n/.Hn�1.M \ @B�/ �Hn�1.†//

and the right side here is exactly C.F.u�/�F.0//, where u�.!/ D��1u.�!/ and F is the area functional of †, and it is possible tocomplete the argument essentially as in the prototypical case.

� � ! IO 12.4 (SLIDE 12/17)


13.1

13 Direct Method Application 2: Structure of singM


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One can partly modify the direct method described above towork in some cases when the n dimensional minimal subman-ifold M has cylindrical tangent cones at some of its singularpoints: i.e. tangent cones C which, after an orthonormal trans-formation of coordinates of the ambient Euclidean space, takethe form C0 � Rm, where C0 is an `-dimensional cone with justthe isolated singularity at 0 (and `Cm D n).Working with such a cylindrical tangent cone setting (startingat some singular point �0 2 M at some scale where M is closeto the cylindrical cone and trying to use Łojasiewicz in the crosssections) works best when there is a singularity with the same,or greater, density than the density of the singularity at �0 ineach cross section (in some cases this is true for topological rea-sons). Neverthess even in the general case approach at leastyields rectifiability results:

Theorem (L.S. 1995): If C is part of a multiplity one classsuch that the m0 is the maximum m such that there existscylindrical cones C D C0 � Rm as above, and if M 2 C thensingM is locally a finite union of locally m0-rectifiable sets.

� � ! IO 13.1 (SLIDE 13/17)


13.2

13 Direct Method Application 2: Structure of singM


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One can partly modify the direct method described above towork in some cases when the n dimensional minimal subman-ifold M has cylindrical tangent cones at some of its singularpoints: i.e. tangent cones C which, after an orthonormal trans-formation of coordinates of the ambient Euclidean space, takethe form C0 � Rm, where C0 is an `-dimensional cone with justthe isolated singularity at 0 (and `Cm D n).Working with such a cylindrical tangent cone setting (startingat some singular point �0 2 M at some scale where M is closeto the cylindrical cone and trying to use Łojasiewicz in the crosssections) works best when there is a singularity with the same,or greater, density than the density of the singularity at �0 ineach cross section (in some cases this is true for topological rea-sons). Neverthess even in the general case approach at leastyields rectifiability results:

Theorem (L.S. 1995): If C is part of a multiplity one classsuch that the m0 is the maximum m such that there existscylindrical cones C D C0 � Rm as above, and if M 2 C thensingM is locally a finite union of locally m0-rectifiable sets.

� � ! IO 13.2 (SLIDE 13/17)


14.1

14 Blowup Methods


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Pioneers in the subject including De Giorgi, Reifenberg, Fed-erer and Almgren were responsible in the 1960’s and 1970’sfor developing blowup methods (involving harmonic approxima-tion) and dimension reducing arguments which proved generalbounds on the size of the possible size of the singular set. Forexample in for codimension 1 area minimizing submanifolds ithas been known since the 1970’s that the singular set has codi-mension at least 7 (and is entirely absent on submanifolds ofdimension � 6); likewise mod 2 minimizers in arbitrary codi-mension at the same time were shown to have codimension atleast 2 and there are many results of this type. But in the mostgeneral situations very little precise information is known be-yond bounds of this kind—almost nothing about the structurebeyond the results mentioned in the previous slides.There are some exceptions though: e.g. In some special classesthe singular sets were completely characterized (e.g. Jean Tay-lor’s work on 2 dimensional soap film minimizers and relatedproblems and Brian White’s work using epiperimetric inequali-ties extends that in some directions) .

� � ! IO 14.1 (SLIDE 14/17)


14.2

14 Blowup Methods


[email protected]


Pioneers in the subject including De Giorgi, Reifenberg, Fed-erer and Almgren were responsible in the 1960’s and 1970’sfor developing blowup methods (involving harmonic approxima-tion) and dimension reducing arguments which proved generalbounds on the size of the possible size of the singular set. Forexample in for codimension 1 area minimizing submanifolds ithas been known since the 1970’s that the singular set has codi-mension at least 7 (and is entirely absent on submanifolds ofdimension � 6); likewise mod 2 minimizers in arbitrary codi-mension at the same time were shown to have codimension atleast 2 and there are many results of this type. But in the mostgeneral situations very little precise information is known be-yond bounds of this kind—almost nothing about the structurebeyond the results mentioned in the previous slides.There are some exceptions though: e.g. In some special classesthe singular sets were completely characterized (e.g. Jean Tay-lor’s work on 2 dimensional soap film minimizers and relatedproblems and Brian White’s work using epiperimetric inequali-ties extends that in some directions) .

� � ! IO 14.2 (SLIDE 14/17)


15.1

15 Asymptotics via Blowup Methods


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In the direction of asymptotics there are some interesting re-sults about minimal graphs—famously the work on extendingthe Bernstein theorem.In this direction we want to describe a relatively recent generalasympotic growth/decay theorem which is proved using blowuptechniques and which has application to lower growth estimatesfor entire solutions of the minimal surface equation.In this theorem we look at multiplicity one classes of surfacesM � Rp (M not necessarily minimal) and we assume that on theregular part of M we are given a non-negative supersolution uof a linear equation: �MuC r�2qu � 0 with q bounded and non-negative. The pair M;q is additionally assumed to be asymptot-ically conic at1 in the sense that for any sequence �k !1 thesequence of “blowdowns” ��1k M converges in the measure theo-retic sense to a (not necessarily unique) cone C, and these conesall have singular sets not too large (Hn�2.C \ K/ < 1 for eachcompact K in fact).

� � ! IO 15.1 (SLIDE 15/17)


15.2



[email protected]


In the direction of asymptotics there are some interesting re-sults about minimal graphs—famously the work on extendingthe Bernstein theorem.In this direction we want to describe a relatively recent generalasympotic growth/decay theorem which is proved using blowuptechniques and which has application to lower growth estimatesfor entire solutions of the minimal surface equation.In this theorem we look at multiplicity one classes of surfacesM � Rp (M not necessarily minimal) and we assume that on theregular part of M we are given a non-negative supersolution uof a linear equation: �MuC r�2qu � 0 with q bounded and non-negative. The pair M;q is additionally assumed to be asymptot-ically conic at1 in the sense that for any sequence �k !1 thesequence of “blowdowns” ��1k M converges in the measure theo-retic sense to a (not necessarily unique) cone C, and these conesall have singular sets not too large (Hn�2.C \ K/ < 1 for eachcompact K in fact).

� � ! IO 15.2 (SLIDE 15/17)


15.3



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We also assume that the correspondingly the sequence qk.x/ Dq.�kx/ converges locally uniformly near points of the regular setof C to a non-negative homogeneous degree zero function qC onq. In case † D reg.C/ \ Sp�1 is connected we let �1.†/ be the“first eigenvalue” of �† � q† using the Rayleigh quotient defini-tion relative to smooth functions on † with compact support in†:

�1.†/ D inf�2C1c .†/;k�k

L2.†/D1

R†.jr�j2 � q†�

2/:

Actually we assume here connectedness of the † to avoid com-plications in the definition of �1.†/, although in fact it is notreally needed.If C is the (compact) class of all cones obtained by the procedureabove then we define finally

�1.M;1/ D supC2C

�1.†/; 0 Dn�22�

q.n�22/2C �1.M;1/;

the latter being interpreted as n�22

in case �1.M;1/ < �.n�22 /2.

Then we obtain the following asymptotic decay theorem: � � ! I

O 15.3 (SLIDE 15/17)


15.4



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Theorem (L.S. 2008): Let u 2 W 1;2loc be a weak non-negative

supersolution of �MuC qu D 0 on regM and < 0. Then foreach p 2 Œ1; n

n�2kukLp.M\B�nB�=2/ � C�

� for sufficiently large �.

Using this theorem one can prove e.g. optimal lower growth es-timates (on approach to1) for exterior solutions of the minimalsurface equation.The proof involves a blowup procedure to compare Lp norms atradius R and �R (� >> 1 fixed).

� � ! IO 15.4 (SLIDE 15/17)

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A Brief Survey of Some Asympotics in the Study of Minimal ...scgas/scgas-2012/Talks/Simon.pdf · divXD R M HXwith sup MjHj