A Brief Survey of Some Asympotics in the Study of Minimal Submanifolds. Leon Simon. Stanford University

Transcription

1 A Brief Survey of Some Asympotics in the Study of Minimal Submanifolds Leon Simon Stanford University

2 Preliminaries Preliminaries Suppose N is some complete Riemannian manifold and M a closed rectifiable subset of N which is minimal (i.e. stationary with respect to the area functional) that is d dt j td0h n.' t.m // D 0 for any family of maps ' t W N! N with the properties that ' t varies smoothly with t 2. 1; 1/, ' 0 D identity and 9 compact K with ' t jn n K D identity on N n K 8t 2. 1; 1/. If N D R p we can take ' t.x/ D x C tx.x/ where X W R p! R p is a smooth function with compact support, and in this case the above stationarity gives the first variation formula R M div M X D 0; where div M means the tangential divergence of X relative to M ; thus div M Xj x D P n id1 i D i Xj x where 1 ; : : : ; n is an orthonormal basis for T x M. (In case the ambient manifold is a N rather than R p we get an identity of the form R M div X D R M H X with sup M jh j < 1.) We could also allow M to have multiplicity, but, except where otherwise explicitly indicated, here we will typically work with multiplicity 1 submanifolds. O 1.1 (SLIDE 1/17)

3 Preliminaries Preliminaries Suppose N is some complete Riemannian manifold and M a closed rectifiable subset of N which is minimal (i.e. stationary with respect to the area functional) that is d dt j td0h n.' t.m // D 0 for any family of maps ' t W N! N with the properties that ' t varies smoothly with t 2. 1; 1/, ' 0 D identity and 9 compact K with ' t jn n K D identity on N n K 8t 2. 1; 1/. If N D R p we can take ' t.x/ D x C tx.x/ where X W R p! R p is a smooth function with compact support, and in this case the above stationarity gives the first variation formula R M div M X D 0; where div M means the tangential divergence of X relative to M ; thus div M Xj x D P n id1 i D i Xj x where 1 ; : : : ; n is an orthonormal basis for T x M. (In case the ambient manifold is a N rather than R p we get an identity of the form R M div X D R M H X with sup M jh j < 1.) We could also allow M to have multiplicity, but, except where otherwise explicitly indicated, here we will typically work with multiplicity 1 submanifolds. O 1.2 (SLIDE 1/17)

4 Preliminaries Preliminaries Suppose N is some complete Riemannian manifold and M a closed rectifiable subset of N which is minimal (i.e. stationary with respect to the area functional) that is d dt j td0h n.' t.m // D 0 for any family of maps ' t W N! N with the properties that ' t varies smoothly with t 2. 1; 1/, ' 0 D identity and 9 compact K with ' t jn n K D identity on N n K 8t 2. 1; 1/. If N D R p we can take ' t.x/ D x C tx.x/ where X W R p! R p is a smooth function with compact support, and in this case the above stationarity gives the first variation formula R M div M X D 0; where div M means the tangential divergence of X relative to M ; thus div M Xj x D P n id1 i D i Xj x where 1 ; : : : ; n is an orthonormal basis for T x M. (In case the ambient manifold is a N rather than R p we get an identity of the form R M div X D R M H X with sup M jh j < 1.) We could also allow M to have multiplicity, but, except where otherwise explicitly indicated, here we will typically work with multiplicity 1 submanifolds. O 1.3 (SLIDE 1/17)

5 Examples Preliminarides (Cont.) Examples Of course there are many examples of such minimal submanifolds. One class which includes many singular examples is the given by the set of complex analytic varieties in C p R 2p for example let M 0 D f 1.0/ where f W C p! C is holomorphic, and let M R 2p be the corresponding real analytic variety. Then (using a calibration argument) one can check that M is area minimizing, so that with n D 2p 2 we have H n.' t.m \ B R // H n.m \ B R /, hence minimal. In this case the singular 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 example quasilinear elliptic PDE theory tells us that we can solve P n i;j D1.1 C the minimal surface equation (i.e. the equation u jduj 2 / 1 D i ud j ud i D j u D 0) on a closed ball B R with arbitrary prescribed continuous boundary data ', and the solution u is continuous on B R and real analytic in MB R D B R R. O 2.1 (SLIDE 2/17)

6 Examples Preliminarides (Cont.) Examples Of course there are many examples of such minimal submanifolds. One class which includes many singular examples is the given by the set of complex analytic varieties in C p R 2p for example let M 0 D f 1.0/ where f W C p! C is holomorphic, and let M R 2p be the corresponding real analytic variety. Then (using a calibration argument) one can check that M is area minimizing, so that with n D 2p 2 we have H n.' t.m \ B R // H n.m \ B R /, hence minimal. In this case the singular 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 example quasilinear elliptic PDE theory tells us that we can solve P n i;j D1.1 C the minimal surface equation (i.e. the equation u jduj 2 / 1 D i ud j ud i D j u D 0) on a closed ball B R with arbitrary prescribed continuous boundary data ', and the solution u is continuous on B R and real analytic in MB R D B R R. O 2.2 (SLIDE 2/17)

7 Examples Preliminarides (Cont.) Examples There are many open questions about the structure of the singular set of singular minimal submanifolds, including some very basic ones: e.g. not known if there can be a minimal M with a sequence of x n of isolated singular points such that x n! p 2 M. Such behavior does not occur in real analytic varieties, so in particular the singular examples discussed on the previous slide cannot not exhibit behavior of this type. Indeed all of the known examples of singular minimal submanifolds have singular sets which stratify into locally finite unions of submanifolds of lower dimension, and it is an open question whether or not such behaviour is generic in some reasonable sense. For instance: can one have an example of a minimal submanifold whose singular set is a (closed) fractional dimensional subset of a straight line or a C 1 curve? Or even simpler questions: can the singular set consist of the union of two disjoint closed subintervals of a straight line in some Euclidean space. O 2.3 (SLIDE 2/17)

8 Examples Preliminarides (Cont.) Examples There are many open questions about the structure of the singular set of singular minimal submanifolds, including some very basic ones: e.g. not known if there can be a minimal M with a sequence of x n of isolated singular points such that x n! p 2 M. Such behavior does not occur in real analytic varieties, so in particular the singular examples discussed on the previous slide cannot not exhibit behavior of this type. Indeed all of the known examples of singular minimal submanifolds have singular sets which stratify into locally finite unions of submanifolds of lower dimension, and it is an open question whether or not such behaviour is generic in some reasonable sense. For instance: can one have an example of a minimal submanifold whose singular set is a (closed) fractional dimensional subset of a straight line or a C 1 curve? Or even simpler questions: can the singular set consist of the union of two disjoint closed subintervals of a straight line in some Euclidean space. O 2.4 (SLIDE 2/17)

9 3 Preliminaries (Cont.): Tangent Cones Tangent Cones 3.1 Aside from its intrinsic interest, an understanding of asymptotic behavior is key to understanding the structure of the singular set General Principle: Sufficiently precise information about asymptotic behavior on approach to singularities ) good information about the structure of the singular set. M always looks asymptotically conic at all sufficiently small scales near a singular point, as we discuss below. The asymptotically conic nature of minimal submanifolds near singular points is of key importance, but its direct usefulness in analyzing singularities is severely limited by the possible nonuniqueness of the limiting cones; i.e. examples that behave analogous to the logarithmic spiral.r/ D re ip j log rj ; 0 < r < 1 (in terms polar coordinates.r; /, D p j log rj, 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 p j log rj C j log j p j log rjj D j log j=.j p j log rj C p j log rj C j log jj/ j log j=j p j log rj! 0 as r # 0 regardless of how small is. O 3.1 (SLIDE 3/17)

10 3 Preliminaries (Cont.): Tangent Cones Tangent Cones 3.2 Aside from its intrinsic interest, an understanding of asymptotic behavior is key to understanding the structure of the singular set General Principle: Sufficiently precise information about asymptotic behavior on approach to singularities ) good information about the structure of the singular set. M always looks asymptotically conic at all sufficiently small scales near a singular point, as we discuss below. The asymptotically conic nature of minimal submanifolds near singular points is of key importance, but its direct usefulness in analyzing singularities is severely limited by the possible nonuniqueness of the limiting cones; i.e. examples that behave analogous to the logarithmic spiral.r/ D re ip j log rj ; 0 < r < 1 (in terms polar coordinates.r; /, D p j log rj, 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 p j log rj C j log j p j log rjj D j log j=.j p j log rj C p j log rj C j log jj/ j log j=j p j log rj! 0 as r # 0 regardless of how small is. O 3.2 (SLIDE 3/17)

11 3 Preliminaries (Cont.): Tangent Cones Tangent Cones 3.3 Aside from its intrinsic interest, an understanding of asymptotic behavior is key to understanding the structure of the singular set General Principle: Sufficiently precise information about asymptotic behavior on approach to singularities ) good information about the structure of the singular set. M always looks asymptotically conic at all sufficiently small scales near a singular point, as we discuss below. The asymptotically conic nature of minimal submanifolds near singular points is of key importance, but its direct usefulness in analyzing singularities is severely limited by the possible nonuniqueness of the limiting cones; i.e. examples that behave analogous to the logarithmic spiral.r/ D re ip j log rj ; 0 < r < 1 (in terms polar coordinates.r; /, D p j log rj, 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 p j log rj C j log j p j log rjj D j log j=.j p j log rj C p j log rj C j log jj/ j log j=j p j log rj! 0 as r # 0 regardless of how small is. O 3.3 (SLIDE 3/17)

12 Monotonicity Preliminaries (Cont.): Monotonicity A key tool in the analysis of asymptotics on approach to a singular point is the monotonicity identity (obtained by taking a radial deformation Xj x D '.jxj/x in the first variation formula). n jm \ B. /j n jm \ B. /j D R M \.B. /nb. // In particular. / D lim #0 n jm \ B. /j exists and n jm \ B. /j. / D R j.x /? j 2 M \B. / jx j nc2 j.x /? j j.x /? j 2. 0/ jx jnc2 In particular R M \.B. /nb. // < 1, which is far from obvious without the help of the above formula, and in fact says jx j nc2 that in some L 1 sense.jx j 1.x //?! 0, which strongly suggests the asymptotically conic nature of M at. (However, to formally check that one has to work a little harder, using a technical variant of the above identity.) O 4.1 (SLIDE 4/17)

13 Monotonicity Preliminaries (Cont.): Monotonicity A key tool in the analysis of asymptotics on approach to a singular point is the monotonicity identity (obtained by taking a radial deformation Xj x D '.jxj/x in the first variation formula). n jm \ B. /j n jm \ B. /j D R M \.B. /nb. // In particular. / D lim #0 n jm \ B. /j exists and n jm \ B. /j. / D R j.x /? j 2 M \B. / jx j nc2 j.x /? j j.x /? j 2. 0/ jx jnc2 In particular R M \.B. /nb. // < 1, which is far from obvious without the help of the above formula, and in fact says jx j nc2 that in some L 1 sense.jx j 1.x //?! 0, which strongly suggests the asymptotically conic nature of M at. (However, to formally check that one has to work a little harder, using a technical variant of the above identity.) O 4.2 (SLIDE 4/17)

14 5 Methods Approaches: A & B 5.1 Here we consider two approaches to the analysis of asymptotic behavior: A. Direct Method Using direct analytic inequalities on the non-linear functional (in particular an infinite dimensional version of some inequalities for the gradient of a real analytic function due to Łojasiewicz) B. Blowup Method Using compactness arguments to prove good approximation to solutions of the relevant non-linear problem by using solutions of the corresponding linearized operator. O 5.1 (SLIDE 5/17)

15 Prototypical example Direct Method Prototype A good prototypical problem for Approach A is the analysis of asympotics at t D 1 for the ODE system./ P D rf./ C R where f is a (real-valued) real analytic on R n 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 analytic varieties, that if 0 is a critical point for f (i.e. rf. 0 / D 0) and f. 0 / D 0, then 9 2.0; 1 and C; > 0 such that 2 (Ł) jrjf j.x/j 2 C; x 2 B. 0 / n {x W f.x/ 0} (Trivial example: If f.x/ D jxj 2 then jrf 1=2 j D jrjxjj 18x.) Claim: Using (Ł) it is easy to prove.t/ has a limit 0 as t! 1 and that j.t/ 0 j C t ˇ for some ˇ > 0 (i.e. power decay asymptotically), provided we assume that remains bounded on the entire interval Œ0; 1/: O 6.1 (SLIDE 6/17)

16 Prototypical example Direct Method Prototype A good prototypical problem for Approach A is the analysis of asympotics at t D 1 for the ODE system./ P D rf./ C R where f is a (real-valued) real analytic on R n 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 analytic varieties, that if 0 is a critical point for f (i.e. rf. 0 / D 0) and f. 0 / D 0, then 9 2.0; 1 and C; > 0 such that 2 (Ł) jrjf j.x/j 2 C; x 2 B. 0 / n {x W f.x/ 0} (Trivial example: If f.x/ D jxj 2 then jrf 1=2 j D jrjxjj 18x.) Claim: Using (Ł) it is easy to prove.t/ has a limit 0 as t! 1 and that j.t/ 0 j C t ˇ for some ˇ > 0 (i.e. power decay asymptotically), provided we assume that remains bounded on the entire interval Œ0; 1/: O 6.2 (SLIDE 6/17)

17 Caution Caution Notice particularly that such a result is false if f is merely smooth rather than real analytic; in the smooth case there are examples 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 that case the set of limit points of.t/ is a circle: Possible picture of curve of steepest descent ( P D rf./) for C 1 function f O 7.1 (SLIDE 7/17)

18 Prototypical ex. (cont.) Direct Method Prototype (Cont.) Checking the Claim: Observe that P.t/. P C R/.1 /jj P 2 so d dt f..t// D.1 / R 1 t jj P 2 f..t// rf..t// P D d dt.f 1..t/// D.1 /f 2./rf./ P D.1 /f 2./rf./.rf./ R/ Cf 2./jrf./j 2 D C jrf =2./j 2 C by (Ł) if.t/ 2 B. 0 /. Integration ) power decay R 1 t jj P 2 1 Cf..t// C t ˇ and hence R 1 t jj P C t 8 t such that.t/ 2 B. 0 /. Integration over intervals Œt 1 ; t 2 ) j.t 2 /.t 1 /j D j R t 2 R P t2 t 1 j t 1 jj P C t if jœt 1 ; t 2 B. 0 /, so if 0 is a limit point of some sequence.t k / (for some sequence t k! 1) then 0 D lim t!1.t/ and j.t/ 0 j C t ˇ=2. (i.e. power decay to the asymptotic limit). 1 ˇ=2 ˇ=2 O 8.1 (SLIDE 8/17)

19 Prototypical ex. (cont.) Direct Method Prototype (Cont.) Checking the Claim: Observe that P.t/. P C R/.1 /jj P 2 so d dt f..t// D.1 / R 1 t jj P 2 f..t// rf..t// P D d dt.f 1..t/// D.1 /f 2./rf./ P D.1 /f 2./rf./.rf./ R/ Cf 2./jrf./j 2 D C jrf =2./j 2 C by (Ł) if.t/ 2 B. 0 /. Integration ) power decay R 1 t jj P 2 1 Cf..t// C t ˇ and hence R 1 t jj P C t 8 t such that.t/ 2 B. 0 /. Integration over intervals Œt 1 ; t 2 ) j.t 2 /.t 1 /j D j R t 2 R P t2 t 1 j t 1 jj P C t if jœt 1 ; t 2 B. 0 /, so if 0 is a limit point of some sequence.t k / (for some sequence t k! 1) then 0 D lim t!1.t/ and j.t/ 0 j C t ˇ=2. (i.e. power decay to the asymptotic limit). 1 ˇ=2 ˇ=2 O 8.2 (SLIDE 8/17)

20 Prototypical ex. (cont.) Direct Method Prototype (Cont.) Checking the Claim: Observe that P.t/. P C R/.1 /jj P 2 so d dt f..t// D.1 / R 1 t jj P 2 f..t// rf..t// P D d dt.f 1..t/// D.1 /f 2./rf./ P D.1 /f 2./rf./.rf./ R/ Cf 2./jrf./j 2 D C jrf =2./j 2 C by (Ł) if.t/ 2 B. 0 /. Integration ) power decay R 1 t jj P 2 1 Cf..t// C t ˇ and hence R 1 t jj P C t 8 t such that.t/ 2 B. 0 /. Integration over intervals Œt 1 ; t 2 ) j.t 2 /.t 1 /j D j R t 2 R P t2 t 1 j t 1 jj P C t if jœt 1 ; t 2 B. 0 /, so if 0 is a limit point of some sequence.t k / (for some sequence t k! 1) then 0 D lim t!1.t/ and j.t/ 0 j C t ˇ=2. (i.e. power decay to the asymptotic limit). 1 ˇ=2 ˇ=2 O 8.3 (SLIDE 8/17)

21 9 Applying the Direct Method to Minimal Submanifolds Realistic Considerations 9.1 Of course the prototypical example above is not completely convincing for 2 reasons: (1) Minimal surfaces, at least away from their singular points, are locally represented as graphs of smooth functions i.e. an infinite dimensional setting, so to generalize from the prototypical case to the actual case we need, at least, a Łojasiewicz inequality in an appropriate infinite dimensional setting. (2) The relevant equations in the infinite dimensional setting (as we ll see below) are more analogous to the second order system R P D rf./ C R rather than P D rf./ C R. It turns out that in fact (2) is not a very serious worry, since a careful analysis shows that the essential difficulties center on the slowly varying case when j R j << j P j, which makes it possible to write R P D rf./ C R in the form P D rf./ C zr where j zrj < z with z a fixed constant in.0; 1/. Indeed using a trick involving a variant of the argument in the prototypical case discussed above (via the monotonicity formula) one can avoid completely any necessity to even address point (2). So (1) is the only problem we need concern ourselves with. O 9.1 (SLIDE 9/17)

22 9 Applying the Direct Method to Minimal Submanifolds Realistic Considerations 9.2 Of course the prototypical example above is not completely convincing for 2 reasons: (1) Minimal surfaces, at least away from their singular points, are locally represented as graphs of smooth functions i.e. an infinite dimensional setting, so to generalize from the prototypical case to the actual case we need, at least, a Łojasiewicz inequality in an appropriate infinite dimensional setting. (2) The relevant equations in the infinite dimensional setting (as we ll see below) are more analogous to the second order system R P D rf./ C R rather than P D rf./ C R. It turns out that in fact (2) is not a very serious worry, since a careful analysis shows that the essential difficulties center on the slowly varying case when j R j << j P j, which makes it possible to write R P D rf./ C R in the form P D rf./ C zr where j zrj < z with z a fixed constant in.0; 1/. Indeed using a trick involving a variant of the argument in the prototypical case discussed above (via the monotonicity formula) one can avoid completely any necessity to even address point (2). So (1) is the only problem we need concern ourselves with. O 9.2 (SLIDE 9/17)

23 Liapunov-Schmidt 10 Liapunov-Schmidt Reduction 10.1 In order to extend the Łojasiewicz inequality to the appropriate infinite dimensional setting, we need an appropriate version of the Liapunov-Schmidt procedure (a procedure which works via the inverse function theorem and enables suitable infinite dimensional problems to be reduced to corresponding finite dimensional ones). We consider a functional F.u/ D R F.!; u; ru/ defined on smooth sections u D u.!/.! 2 / of the normal bundle over, with a smooth compact Riemannian manifold of dimension m 1, and F is assumed to be smooth in its dependence on!; u; ru. (We ll need to assume that F is in fact real analytic in its dependence on u; ru in the discussion of the Łojasiewicz inequality for F below.) M.u/ (or rf.u/ ) denotes the (second order) Euler-Lagrange operator (or first variation operator ) of F characterized by hm.u/; i L 2 D d ds j sd0f.u C s/: O 10.1 (SLIDE 10/17)

24 Liapunov-Schmidt 10 Liapunov-Schmidt Reduction 10.2 In order to extend the Łojasiewicz inequality to the appropriate infinite dimensional setting, we need an appropriate version of the Liapunov-Schmidt procedure (a procedure which works via the inverse function theorem and enables suitable infinite dimensional problems to be reduced to corresponding finite dimensional ones). We consider a functional F.u/ D R F.!; u; ru/ defined on smooth sections u D u.!/.! 2 / of the normal bundle over, with a smooth compact Riemannian manifold of dimension m 1, and F is assumed to be smooth in its dependence on!; u; ru. (We ll need to assume that F is in fact real analytic in its dependence on u; ru in the discussion of the Łojasiewicz inequality for F below.) M.u/ (or rf.u/ ) denotes the (second order) Euler-Lagrange operator (or first variation operator ) of F characterized by hm.u/; i L 2 D d ds j sd0f.u C s/: O 10.2 (SLIDE 10/17)

25 Liapunov-Schmidt 10 Liapunov-Schmidt Reduction 10.3 In order to extend the Łojasiewicz inequality to the appropriate infinite dimensional setting, we need an appropriate version of the Liapunov-Schmidt procedure (a procedure which works via the inverse function theorem and enables suitable infinite dimensional problems to be reduced to corresponding finite dimensional ones). We consider a functional F.u/ D R F.!; u; ru/ defined on smooth sections u D u.!/.! 2 / of the normal bundle over, with a smooth compact Riemannian manifold of dimension m 1, and F is assumed to be smooth in its dependence on!; u; ru. (We ll need to assume that F is in fact real analytic in its dependence on u; ru in the discussion of the Łojasiewicz inequality for F below.) M.u/ (or rf.u/ ) denotes the (second order) Euler-Lagrange operator (or first variation operator ) of F characterized by hm.u/; i L 2 D d ds j sd0f.u C s/: O 10.3 (SLIDE 10/17)

26 10 Liapunov-Schmidt Reduction Liapunov-Schmidt 10.4 L.u/ D d ds j sd0m.su/ is the linearization of M at 0, assumed to be elliptic, so that L D kernel L is finite dimensional, with orthonormal basis ' 1 ; : : : ; ' q (relative to L 2. /), hence P L.u/ D P q j D1 j ' j. j D hu; ' j i L 2. // is the orthogonal projection onto L. Then N.u/ D P L.u/ C M.u/ has trivial kernel, so N has a smooth (or real-analytic if F is real analytic in its dependence on u; ru) inverse defined in a nhd. of 0. Then.N.u// u and hence M.u/ D 0 u D.P L.u// u 2 M in a suitable nhd. of 0, where M is the smooth (or real analytic) manifold of dimension q. O 10.4 (SLIDE 10/17)

27 10 Liapunov-Schmidt Reduction 10.5 Thus we have the following elegant and very useful picture: Liapunov-Schmidt Liapunov-Schmidt Reduction The blue curves are contained in the manifold M, and represent the set of all possible solutions of the non-linear equation M.u/ D 0 for small u. Some 1-variable calculus computations along 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.. P n j D1 j ' j // O 10.5 (SLIDE 10/17)

28 10 Liapunov-Schmidt Reduction 10.6 and in addition Liapunov-Schmidt C 1 km.u/k L 2. / jrf./j C km.u/k L 2. / jf.u/ f./j C km.u/k 2 L 2. / ; for all sufficiently small u, where we continue to use the notation that D. 1 ; : : : ; q / 2 R q with j D hu; ' j i L 2. /. O 10.6 (SLIDE 10/17)

29 Łojasiewicz in dim 1 11 Infinite Dimensional Łojasiewicz Inequality 11.1 As mentioned above, if F (the integrand in the functional F) has real analytic dependence on u; ru then f./ is a real analytic function of, so we can use the Łojasiewicz inequality: jf./ f.0/j 1 C jrf./j; jj < ; for suitable 2.0; 1 and C; > 0, and then the last two inequalities on the previous slide 2 imply jf.u/ F.0/j 1 C km.u/k L 2. / for u in a small enough (C 3 ) neighborhood of 0 (with the same but slightly larger C ). This is the infinite dimensional version of the Łojasiewicz inequality. In the applications to minimal submanifolds discussed below, we use the special case D C\S N, where C is an n-dimensional minimal cone in R N C1 with vertex at 0 and no other singularities, 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 H n 1.graph u/ O 11.1 (SLIDE 11/17)

30 Łojasiewicz in dim 1 11 Infinite Dimensional Łojasiewicz Inequality 11.2 As mentioned above, if F (the integrand in the functional F) has real analytic dependence on u; ru then f./ is a real analytic function of, so we can use the Łojasiewicz inequality: jf./ f.0/j 1 C jrf./j; jj < ; for suitable 2.0; 1 and C; > 0, and then the last two inequalities on the previous slide 2 imply jf.u/ F.0/j 1 C km.u/k L 2. / for u in a small enough (C 3 ) neighborhood of 0 (with the same but slightly larger C ). This is the infinite dimensional version of the Łojasiewicz inequality. In the applications to minimal submanifolds discussed below, we use the special case D C\S N, where C is an n-dimensional minimal cone in R N C1 with vertex at 0 and no other singularities, 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 H n 1.graph u/ O 11.2 (SLIDE 11/17)

31 Direct Method App I 12 Uniqueness of Tangent Cone Theorem via Direct Method 12.1 Theorem. If M is an n-dimensional minimal submanifold of R p with 0 2 sing M, and if C is one of the tangent cones of M at 0 (so that k 1 M converges, in the appropriate weak measure sense, to C for some sequence k # 0, then C is the unique tangent cone, the singularity of M at 0 is isolated and M is asymptotic to C at rate C jxj ˇ. To prove this pick a starting value > 0 so that 1 M is measure theoretically close (within some arbitrary " > 0) to C in an annular region < r D jxj < 1. Now using the Allard regularity theorem 1 M must be close to C in the C 2 sense in the smaller annular region 2 < r < 1 in the sense that 2 M \ {x W 2 < jxj < 1} 2 D graph u with u a C 2 section of small norm of the normal bundle over C \ {x W 2 < jxj < 1 }, and with 2 u satisfying the Minimal Surface Equation M C.u/ D 0 (Euler- Lagrange equation for the area functional over C). O 12.1 (SLIDE 12/17)

32 Direct Method App I 12 Uniqueness of Tangent Cone Theorem via Direct Method 12.2 Theorem. If M is an n-dimensional minimal submanifold of R p with 0 2 sing M, and if C is one of the tangent cones of M at 0 (so that k 1 M converges, in the appropriate weak measure sense, to C for some sequence k # 0, then C is the unique tangent cone, the singularity of M at 0 is isolated and M is asymptotic to C at rate C jxj ˇ. To prove this pick a starting value > 0 so that 1 M is measure theoretically close (within some arbitrary " > 0) to C in an annular region < r D jxj < 1. Now using the Allard regularity theorem 1 M must be close to C in the C 2 sense in the smaller annular region 2 < r < 1 in the sense that 2 M \ {x W 2 < jxj < 1} 2 D graph u with u a C 2 section of small norm of the normal bundle over C \ {x W 2 < jxj < 1 }, and with 2 u satisfying the Minimal Surface Equation M C.u/ D 0 (Euler- Lagrange equation for the area functional over C). O 12.2 (SLIDE 12/17)

33 Direct Method App I 12 Uniqueness of Tangent Cone Theorem via Direct Method 12.3 In terms of coordinates t D log r and! D x=jxj 2 D C \ S p 1 one can check that M C.u/ D 0 can be written in the form Ru n Pu D M.u/ C R where M is the Euler-Lagrange operator for the area functional on (so the infinite dimensional version of the Łojasiewicz 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 a slight variant of the monotonicity formula and the infinite dimensional version of the Łojasiewicz inequality. (Still very reminiscent of the argument we used in the prototypical case.) The variant of the monotonicity formula mentioned above is that (by extra step in the previous monotonicity discussion) R j.x /? j 2 M \B. / C.n/.H n 1.M jx j nc2 / H n 1. // and the right side here is exactly C.F.u / F.0//, where u.!/ D 1 u.!/ and F is the area functional of, and it is possible to complete the argument essentially as in the prototypical case. O 12.3 (SLIDE 12/17)

34 Direct Method App I 12 Uniqueness of Tangent Cone Theorem via Direct Method 12.4 In terms of coordinates t D log r and! D x=jxj 2 D C \ S p 1 one can check that M C.u/ D 0 can be written in the form Ru n Pu D M.u/ C R where M is the Euler-Lagrange operator for the area functional on (so the infinite dimensional version of the Łojasiewicz 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 a slight variant of the monotonicity formula and the infinite dimensional version of the Łojasiewicz inequality. (Still very reminiscent of the argument we used in the prototypical case.) The variant of the monotonicity formula mentioned above is that (by extra step in the previous monotonicity discussion) R j.x /? j 2 M \B. / C.n/.H n 1.M jx j nc2 / H n 1. // and the right side here is exactly C.F.u / F.0//, where u.!/ D 1 u.!/ and F is the area functional of, and it is possible to complete the argument essentially as in the prototypical case. O 12.4 (SLIDE 12/17)

35 13 Direct Method Application 2: Structure of sing M Direct Method App II 13.1 One can partly modify the direct method described above to work in some cases when the n dimensional minimal submanifold M has cylindrical tangent cones at some of its singular points: i.e. tangent cones C which, after an orthonormal transformation of coordinates of the ambient Euclidean space, take the form C 0 R m, where C 0 is an `-dimensional cone with just the isolated singularity at 0 (and ` C m D n). Working with such a cylindrical tangent cone setting (starting at some singular point 0 2 M at some scale where M is close to the cylindrical cone and trying to use Łojasiewicz in the cross sections) works best when there is a singularity with the same, or greater, density than the density of the singularity at 0 in each cross section (in some cases this is true for topological reasons). Neverthess even in the general case approach at least yields rectifiability results: Theorem (L.S. 1995): If C is part of a multiplity one class such that the m 0 is the maximum m such that there exists cylindrical cones C D C 0 R m as above, and if M 2 C then sing M is locally a finite union of locally m 0 -rectifiable sets. O 13.1 (SLIDE 13/17)

36 13 Direct Method Application 2: Structure of sing M Direct Method App II 13.2 One can partly modify the direct method described above to work in some cases when the n dimensional minimal submanifold M has cylindrical tangent cones at some of its singular points: i.e. tangent cones C which, after an orthonormal transformation of coordinates of the ambient Euclidean space, take the form C 0 R m, where C 0 is an `-dimensional cone with just the isolated singularity at 0 (and ` C m D n). Working with such a cylindrical tangent cone setting (starting at some singular point 0 2 M at some scale where M is close to the cylindrical cone and trying to use Łojasiewicz in the cross sections) works best when there is a singularity with the same, or greater, density than the density of the singularity at 0 in each cross section (in some cases this is true for topological reasons). Neverthess even in the general case approach at least yields rectifiability results: Theorem (L.S. 1995): If C is part of a multiplity one class such that the m 0 is the maximum m such that there exists cylindrical cones C D C 0 R m as above, and if M 2 C then sing M is locally a finite union of locally m 0 -rectifiable sets. O 13.2 (SLIDE 13/17)

37 Blowup Methods 14 Blowup Methods 14.1 Pioneers in the subject including De Giorgi, Reifenberg, Federer and Almgren were responsible in the 1960 s and 1970 s for developing blowup methods (involving harmonic approximation) and dimension reducing arguments which proved general bounds on the size of the possible size of the singular set. For example in for codimension 1 area minimizing submanifolds it has been known since the 1970 s that the singular set has codimension at least 7 (and is entirely absent on submanifolds of dimension 6); likewise mod 2 minimizers in arbitrary codimension at the same time were shown to have codimension at least 2 and there are many results of this type. But in the most general situations very little precise information is known beyond bounds of this kind almost nothing about the structure beyond the results mentioned in the previous slides. There are some exceptions though: e.g. In some special classes the singular sets were completely characterized (e.g. Jean Taylor s work on 2 dimensional soap film minimizers and related problems and Brian White s work using epiperimetric inequalities extends that in some directions). O 14.1 (SLIDE 14/17)

38 Blowup Methods 14 Blowup Methods 14.2 Pioneers in the subject including De Giorgi, Reifenberg, Federer and Almgren were responsible in the 1960 s and 1970 s for developing blowup methods (involving harmonic approximation) and dimension reducing arguments which proved general bounds on the size of the possible size of the singular set. For example in for codimension 1 area minimizing submanifolds it has been known since the 1970 s that the singular set has codimension at least 7 (and is entirely absent on submanifolds of dimension 6); likewise mod 2 minimizers in arbitrary codimension at the same time were shown to have codimension at least 2 and there are many results of this type. But in the most general situations very little precise information is known beyond bounds of this kind almost nothing about the structure beyond the results mentioned in the previous slides. There are some exceptions though: e.g. In some special classes the singular sets were completely characterized (e.g. Jean Taylor s work on 2 dimensional soap film minimizers and related problems and Brian White s work using epiperimetric inequalities extends that in some directions). O 14.2 (SLIDE 14/17)

39 General Decay Lem. 15 Asymptotics via Blowup Methods 15.1 In the direction of asymptotics there are some interesting results about minimal graphs famously the work on extending the Bernstein theorem. In this direction we want to describe a relatively recent general asympotic growth/decay theorem which is proved using blowup techniques and which has application to lower growth estimates for entire solutions of the minimal surface equation. In this theorem we look at multiplicity one classes of surfaces M R p (M not necessarily minimal) and we assume that on the regular part of M we are given a non-negative supersolution u of a linear equation: M u C r 2 qu 0 with q bounded and nonnegative. The pair M; q is additionally assumed to be asymptotically conic at 1 in the sense that for any sequence k! 1 the sequence of blowdowns k 1 M converges in the measure theoretic sense to a (not necessarily unique) cone C, and these cones all have singular sets not too large (H n 2.C \ K/ < 1 for each compact K in fact). O 15.1 (SLIDE 15/17)

40 General Decay Lem. 15 Asymptotics via Blowup Methods 15.2 In the direction of asymptotics there are some interesting results about minimal graphs famously the work on extending the Bernstein theorem. In this direction we want to describe a relatively recent general asympotic growth/decay theorem which is proved using blowup techniques and which has application to lower growth estimates for entire solutions of the minimal surface equation. In this theorem we look at multiplicity one classes of surfaces M R p (M not necessarily minimal) and we assume that on the regular part of M we are given a non-negative supersolution u of a linear equation: M u C r 2 qu 0 with q bounded and nonnegative. The pair M; q is additionally assumed to be asymptotically conic at 1 in the sense that for any sequence k! 1 the sequence of blowdowns k 1 M converges in the measure theoretic sense to a (not necessarily unique) cone C, and these cones all have singular sets not too large (H n 2.C \ K/ < 1 for each compact K in fact). O 15.2 (SLIDE 15/17)

41 General Decay Lem. 15 Asymptotics via Blowup Methods 15.3 We also assume that the correspondingly the sequence q k.x/ D q. k x/ converges locally uniformly near points of the regular set of C to a non-negative homogeneous degree zero function q C on q. In case D reg.c/ \ S p 1 is connected we let 1. / be the first eigenvalue of q using the Rayleigh quotient definition relative to smooth functions on with compact support in : R 1. / D inf.jrj2 q 2 /: 2C 1 c. /;kk L 2. / D1 Actually we assume here connectedness of the to avoid complications in the definition of 1. /, although in fact it is not really needed. If C is the (compact) class of all cones obtained by the procedure above then we define finally q 1.M; 1/ D sup 1. /; 0 D n 2. n /2 C 1.M; 1/; C2C the latter being interpreted as n 2 2 in case 1.M; 1/ <. n 2 2 /2. Then we obtain the following asymptotic decay theorem: O 15.3 (SLIDE 15/17)

42 15 Asymptotics via Blowup Methods 15.4 General Decay Lem. Theorem (L.S. 2008): Let u 2 W 1;2 loc be a weak non-negative supersolution of M u C qu D 0 on reg M and < 0. Then for n each p 2 Œ1; n 2 kuk L p.m \B nb =2 / C for sufficiently large. Using this theorem one can prove e.g. optimal lower growth estimates (on approach to 1) for exterior solutions of the minimal surface equation. The proof involves a blowup procedure to compare L p norms at radius R and R ( >> 1 fixed). O 15.4 (SLIDE 15/17)