A Remark on the Non-Compactness of W 2,d Immersions of d -Dimensional Hypersurfaces
aa r X i v : . [ m a t h . DG ] J un A REMARK ON THE NON-COMPACTNESS OF W ,d -IMMERSIONS OF d -DIMENSIONAL HYPERSURFACES SIRAN LI
Abstract.
We consider the continuous W ,d -immersions of d -dimensional hypersurfaces in R d +1 with second fundamental forms uniformly bounded in L d . Two results are obtained: first,we construct a family of such immersions whose limit fails to be an immersion of a manifold.This addresses the endpoint cases in J. Langer [6] and P. Breuning [1]. Second, under theadditional assumption that the Gauss map is slowly oscillating, we prove that any family ofsuch immersions subsequentially converges to a set locally parametrised by Hölder functions. Introduction
In [6], motivated by J. Cheeger’s finiteness theorems ([2], also see K. Corlette [5]) and theWillmore energy of surfaces (see e.g.
T. Rivière [7]), J. Langer proved the following result:
Let
A, E are given finite numbers and p > . Denote by F ( A, E, p ) the moduli space ofimmersed surfaces ψ : M → R with Area( ψ ) ≤ A , k II k L p ( M ) ≤ E and R M ψ d V = 0 . Thenany sequence { ψ j } ⊂ F ( A, E, p ) contains a subsequence converging in C to an immersed surfacemodulo Diff( M ) , the group of diffeomorphisms of M .Here and hereafter, the immersed submanifold ψ : M ֒ → R n is equipped with the pullbackmetric from the Euclidean metric on R n . We denote by II the second fundamental form of ψ ,and we write d V for the volume form on M .In a recent paper [1], P. Breuning generalised J. Langer’s result to arbitrary dimensionsand co-dimensions: Let
A, E be given finite numbers and p > n , d > n . Denote by F ( V, E, d, n ) the modulispace of immersions ψ : M → R n where M is a d -dimensional closed manifold, Vol( M ) ≤ A , k II k L p ( M ) ≤ E and ψ ( M ) contains a fixed point. Then any sequence { ψ j } ⊂ F ( V, E, d, n ) contains a subsequence converging in C to an immersed submanifold modulo Diff( M ) . The above two compactness theorems on the moduli space of immersions have a crucialassumption: p > dim( M ) = d . Indeed, the proofs in [6, 1] utilise the Sobolev–Morrey embedding W ,p ( R n ) ֒ → C ,α ( R n ) , where p > n and α = α ( p, n ) ∈ ]0 , . It is natural to ask about the end-point case p = d , for which the Sobolev–Morrey embedding fails. In the case p = d = 2 , J. Langer(p.227, [6]) constructed a counterexample using conformal geometry — the Möbius inversions ofthe Clifford torus T cl with respect to a sequence of points x j / ∈ T cl approaching an outermostpoint (with distance measured from the centre of the embedded image of T cl ) on T cl cannot tendto any immersed manifold. It crucially relies on the structure of C . Date : June 11, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Immersions; Hypersurface; Chord-Arc Surface; Second Fundamental Form; Gauss Map;Compactness; Bounded Mean Oscillations (BMO); Finiteness Theorems; Riemannian Geometry. ur first goal of this paper is to construct a counterexample for p = d in arbitrary di-mensions. The idea is to construct a family of hypersurfaces “spiralling wildly”, as a vaguereminiscence of the motion of vortex sheets in fluid dynamics. This is achieved by letting theGauss map n ( i.e. , the outward unit normal vectorfield) take N ≫ turns in one direction as weapproach some fixed point O and, symmetrically, take N turns in the opposite direction as weleave O . To illustrate the geometric picture, we first discuss the case d = 1 , and then constructa counterexample for general d . Instead of using conformal geometric methods, we exploit thescaling invariance of k II k L d ( M ) , which holds in arbitrary dimensions. This is the content of § .Our second goal is to establish an affirmative compactness result for the p = d case, withthe help of an additional hypothesis: the BM O -norm of the Gauss map n , k n k BMO( M ) := sup x ∈M , R> − Z M∩ B ( x,R ) | n ( y ) − n x,R | d V ( y ) , (1.1)is small. Throughout B ( x, R ) denotes the geodesic ball of radius R and centre x in M , − R is theaveraged integral, and n x,R := − R B ( x,R ) n d V . This is inspired by the works [8, 9, 10] due to S.Semmes on chord-arc surfaces with small constants . In § we use the results in [8, 9, 10] to provea “partial regularity” theorem for the weak limit: given any family of immersed hypersurfaces R d in the ( d +1) -dimensional Euclidean space with uniformly L d -bounded second fundamental formsand small k n k BMO( M ) , one may extract a subsequence whose limit can be locally parametrisedby Hölder functions.The paper is concluded by several further remarks in § .2. A counter-example to the endpoint case p = d Let us first study the toy model d = 1 . We prove the following simple result: Lemma 2.1.
There exist a family of smooth curves {M ǫ } each homeomorphic to R , and afamily of immersions ψ ǫ : M ǫ → R as planar curves, such that the extrinsic curvatures { II ǫ } associated to { ψ ǫ } are uniformly bounded in L , but { ψ ǫ ◦ σ ǫ } does not converge in C -topologyto any immersion of R for arbitrary { σ ǫ } ⊂ Diff( R ) . The extrinsic curvature of a planar curve is the mean curvature. Recall that the meancurvature is defined in arbitrary dimensions as the trace of the second fundamental form. In thecase d = 1 we may still denote the extrinsic curvature by II . Proof.
Let J ∈ C ∞ c ( R ) be a standard symmetric mollifier; e.g. , J ( s ) := Λ exp ® s − ´ {| s | < } , (2.1)where the universal constant Λ > is chosen such that R R J ( s ) d s = 1 . As usual J ǫ ( s ) := ǫ − J ( s/ǫ ) for ǫ > ; then k J ǫ k L ( R ) = 1 for every ǫ > . In addition, define the kernel K ǫ ( x ) := J ǫ ( x + ǫ ) − J ǫ ( x − ǫ ) . (2.2)It satisfies k K ǫ k L ( R ) = 2 , K ǫ ∈ C ∞ c ( R ) and spt( K ǫ ) = [ − ǫ, ǫ ] ; in particular, it is smooth at .Now, define an angle function θ ǫ ( x ) := 10 m · π Z x −∞ K ǫ ( s ) d s, (2.3) here m ∈ Z + is to be determined. Then, choose the Gauss map n ǫ ∈ C ∞ ( R ; S ) by n ǫ ( x ) := " cos θ ǫ ( x )sin θ ǫ ( x ) for each x ∈ R . (2.4)The extrinsic curvature II ǫ equals to the negative of the gradient of the Gauss map; hence | II ǫ ( x ) | = … (cid:12)(cid:12)(cid:12) Ä − sin θ ǫ ( x ) ä ( θ ǫ ) ′ ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Ä cos θ ǫ ( x ) ä ( θ ǫ ) ′ ( x ) (cid:12)(cid:12)(cid:12) = | ( θ ǫ ) ′ ( x ) | = (2 π · m ) K ǫ ( x ) . (2.5)Thus, the L norm of { II ǫ } is uniformly bounded by π · m .Let ψ ǫ be a smooth immersion that realises the Gauss map n ǫ whose image is the unitcircle S in R . For each η > , we may easily modify ψ ǫ to ˜ ψ ǫ such that | ˜ ψ ǫ ( x ) | is decreasingon ] − ∞ , and increasing on [0 , ∞ [ , the image of ˜ ψ ǫ in R is homeomorphic to R , and that k ψ ǫ − ˜ ψ ǫ k C ( R ) < η. (2.6)Indeed, notice that the image of ψ ǫ (cid:12)(cid:12)(cid:12) ] − ∞ , covers S for m times in the positive orientation,and the image of ψ ǫ (cid:12)(cid:12)(cid:12) [0 , ∞ [ covers S for m times in the negative orientation. We then choosethe perturbed map ˜ ψ ǫ such that • As x goes from −∞ to , ˜ ψ ǫ wraps around the origin in a helical trajectory for m times.Moreover, in each round | ˜ ψ ǫ | decreases monotonically by ∼ − m ; • As x increases from to ∞ , ˜ ψ ǫ “unwraps” around the origin along a helix for m times,in each round | ˜ ψ ǫ | increases monotonically by ∼ − m ; • For x ∈ ] − ∞ , − ǫ ] ⊔ [2 ǫ, + ∞ [ , the image of ˜ ψ ǫ consists of straight line segments (“longflat tails”); hence n ǫ stays constant on each component of ] − ∞ , − ǫ ] ⊔ [2 ǫ, + ∞ [ ; • Finally, the image ˜ ψ ǫ ( R ) is C ∞ and homeomorphic to R .In view of the above properties, one can take m = m ( η ) ∈ Z + sufficiently large to verify (2.6).Let us pick η = , so m is a universal constant fixed once and for all. Without loss of generality,from now on we may assume ψ ǫ = ˜ ψ ǫ . The point is to ensure that the image of ψ ǫ in R is free ofloops and “concentrates” near the origin ∈ R , with Gauss map and second fundamental formarbitrarily staying close to those constructed in (2.4) and (2.5), respectively.To conclude the proof, let us define M ǫ as the homeomorphic copy of R equipped withthe pullback metric ( ψ ǫ ) δ ij , where δ ij is the Euclidean metric on the ambient space R . Itremains to show that the C -limit (modulo Diff( R ) ) of ψ ǫ as ǫ → + cannot be an immersion.Indeed, note that the topological degree satisfies deg (cid:16) ψ ǫ (cid:12)(cid:12)(cid:12) ] − ∞ , (cid:17) = 10 m , deg (cid:16) ψ ǫ (cid:12)(cid:12)(cid:12) [0 , ∞ [ (cid:17) = − m . (2.7)These identities are independent of ǫ . Hence, if ¯ ψ were a limiting immersion, (2.7) would havebeen preserved. However, K ǫ ∗ ⇀ δ − δ = 0 as measures, so (2.3)(2.4)(2.5) imply that anypointwise subsequential limit of ψ ǫ has zero topological degree. Hence we get the contradictionand the proof is complete. (cid:3) Three remarks are in order: From (2.5) one may infer that k II ǫ k L ∞ ( M ǫ ) = 2 π · m · Λ eǫ + η −→ ∞ as ǫ → + . . The construction in Lemma 2.1 can be localised near . We can restrict M ǫ to curvesof finite H measure by removing the long tails. This recovers the volume bounds in [6, 1] ( § ). We can construct φ ǫ whose limit blows up at a countable discrete set { x n } by taking ˜ θ ǫ ( x ) := ∞ X n =1 − n B ( x n ,R n ) ( x ) θ ǫ ( x ) in place of θ ǫ ( x ) , where { B ( x n , R n ) } are disjoint for all n . Geometrically, the immersed imagescorresponding to ˜ θ ǫ are smooth curves that spiral towards the centres x n when x < x n , andthen spiral away from x n when x > x n . Near x n the rate of motion blows up in L ∞ as ǫ → + ;nevertheless, its L norm is constant.Now let us generalise the above construction to d -dimensions: Theorem 2.2.
Let d ≥ be an integer. There exist a family of smooth manifolds {M ǫ } eachhomeomorphic to R d , and a family of immersions ψ ǫ : M ǫ → R d +1 as smooth hypersurfaces,such that the second fundamental forms { II ǫ } associated to { ψ ǫ } are uniformly bounded in L d , but { ψ ǫ ◦ σ ǫ } does not converge in C -topology to any immersion of R d for arbitrary { σ ǫ } ⊂ Diff( R d ) .Proof. Again the crucial point is to construct the Gauss map n ǫ ∈ C ∞ ( R d ; S d ) . We make use ofthe spherical coordinates on S d . For x = ( x , x , . . . , x d ) ∈ R d , one needs to specify the anglefunctions θ ǫi : R d → [0 , π [ in the following: n ǫ ( x ) = cos θ ǫ ( x )sin θ ǫ ( x ) cos θ ǫ ( x )sin θ ǫ ( x ) sin θ ǫ ( x ) cos θ ǫ ( x ) ... sin θ ǫ ( x ) · · · sin θ ǫd − ( x ) cos θ ǫd ( x )sin θ ǫ ( x ) · · · sin θ ǫd − ( x ) sin θ ǫd ( x ) . (2.8)Throughout S d = { z ∈ R d +1 : | z | = 1 } is the round sphere.Indeed, let us choose θ ǫi ( x ) ≡ Θ ǫ ( x i ) := 10 m · π Z x i −∞ K ǫ ( s ) d s, (2.9)where the kernel K ǫ is defined as in (2.2), and m ∈ Z + is a large universal constant to be fixedlater. Each θ ǫi is a function of x i only. One can easily compute all the entries in − II ǫ = ∇ n ǫ ,which is a lower-triangular d × ( d + 1) matrix due to the embedding S d ֒ → R d +1 . The rows { r i } i =1 , ,...,d of {∇ n ǫ } are: r = (cid:16) − (Θ ǫ ) ′ ( x ) sin Θ ǫ ( x ) , , · · · , (cid:17) , r = (cid:16) (Θ ǫ ) ′ ( x ) cos Θ ǫ ( x ) cos Θ ǫ ( x ) , − (Θ ǫ ) ′ ( x ) sin Θ ǫ ( x ) sin Θ ǫ ( x ) , , · · · , (cid:17) , r = (cid:16) (Θ ǫ ) ′ ( x ) sin Θ ǫ ( x ) sin Θ ǫ ( x ) cos Θ ǫ ( x ) , (Θ ǫ ) ′ ( x ) sin Θ ǫ ( x ) cos Θ ǫ ( x ) cos Θ ǫ ( x ) , − (Θ ǫ ) ′ ( x ) sin Θ ǫ ( x ) sin Θ ǫ ( x ) sin Θ ǫ ( x ) , , · · · , (cid:17) so on and so forth, with the last two being r d − = (cid:16) (Θ ǫ ) ′ ( x ) cos Θ ǫ ( x ) sin Θ ǫ ( x ) · · · sin Θ ǫ ( x d − ) cos Θ ǫ ( x d ) , ⋆, · · · , ⋆, Θ ǫ ) ′ ( x d − ) sin Θ ǫ ( x ) sin Θ ǫ ( x ) · · · cos Θ ǫ ( x d − ) cos Θ ǫ ( x d ) , − (Θ ǫ ) ′ ( x d ) sin Θ ǫ ( x ) sin Θ ǫ ( x ) · · · sin Θ ǫ ( x d − ) sin Θ ǫ ( x d ) (cid:17) and r d = (cid:16) (Θ ǫ ) ′ ( x ) cos Θ ǫ ( x ) sin Θ ǫ ( x ) · · · sin Θ ǫ ( x d − ) sin Θ ǫ ( x d ) , ⋆, · · · , ⋆, (Θ ǫ ) ′ ( x d − ) sin Θ ǫ ( x ) sin Θ ǫ ( x ) · · · cos Θ ǫ ( x d − ) sin Θ ǫ ( x d ) , (Θ ǫ ) ′ ( x d ) sin Θ ǫ ( x ) sin Θ ǫ ( x ) · · · sin Θ ǫ ( x d − ) cos Θ ǫ ( x d ) (cid:17) . A direct computation yields the Hilbert–Schmidt norm of the second fundamental form: | II ǫ | = |∇ n ǫ | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) (Θ ǫ ) ′ ( x ) , · · · , (Θ ǫ ) ′ ( x d ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . (2.10)Thus, in view of (2.9) and Fubini’s theorem, we have k II ǫ k L d ( R d ) = 10 m · π (cid:13)(cid:13)(cid:13)(cid:13) K ǫ ⊗ · · · ⊗ K ǫ | {z } d times (cid:13)(cid:13)(cid:13)(cid:13) L d ( R d ) = 10 m · π k K ǫ k L ( R d ) = 10 m · π. (2.11)It now remains to choose a smooth immersion that realises n ǫ (approximately). Theconstruction is similar to Lemma 2.1 in the case of d = 1 . First, take ψ ǫ whose Gauss mapis n ǫ . Geometrically, ψ ǫ winds around S d — accelerating on the first half and decelerating onthe second half of the trajectory — with respect to a given orientation for m “cycles”, andthen undoes the winding by turning symmetrically in the opposite orientation. In the above, by“cycle” we mean a generator of the cohomology group H d ( S d ) ∼ = R .In what follows we shall describe how to modify the above construction to obtain a coun-terexample with M ǫ homeomorphic to R d . We shall construct ˜ ψ ǫ , modified versions of ψ ǫ , suchthat for each ǫ > the image of ˜ ψ ǫ in R d +1 is a smooth homeomorphic copy of R d . In addi-tion, each such image has flat ends outside B (0 , and has d independent angular variables inthe spherical coordinates, i.e. , the Gauss map still takes the form (2.8) with θ ǫi ( x ) ≡ Θ ǫ ( x i ) .Moreover, such ˜ ψ ǫ differs from ψ ǫ only by an arbitrarily small error in the C -topology.To obtain the modified immersions ˜ ψ ǫ , let us begin with S ǫ ≡ image of ψ ǫ . For each ǫ > , schematically we can write S ǫ ∼ +10 m S d − m S d , with ± denoting the orientation. For x / ∈ B (0 , we have ψ ǫ ≡ , in view of (2.9) and the choice of K ǫ . Now, for some small number < η ≪ − m to be specified, we shall modify S ǫ as follows.First of all, for the concentric spheres S d and (1 − η ) S d := ∂B (0 , − η ) in R d +1 , we cansmoothly “interpolate” between them by finding a hypersurface S ǫ lying in the annulus formedby the two spheres, such that the tangent spaces of S ǫ and S ǫ coincide at the “north poles” e and (1 − η ) e , and that all the angular variables θ , . . . θ d on S ǫ change by π at the same constantspeed. Here and hereafter e := (0 , . . . , , . In the same way, we construct S ǫ nested between (1 − η ) S d and (1 − η ) S d , so that S ǫ and S ǫ can be glued smoothly at (1 − η ) e , and their naturalorientations are the same. Let us repeat this process to get S ǫj for j ∈ { , , . . . , m } , and glue m [ j =1 S ǫj := S ǫ + to form a smooth “spiral” starting from e and ending at (1 − m η ) e . o proceed, denote by S ǫ − the hypersurface obtained via shifting S ǫ + to its right-hand sideby η/ . This is well-defined as S ǫ + an oriented hypersurface in R d +1 , and we have S ǫ + ∩ S ǫ − = ∅ .Let us endow S ǫ − with the orientation opposite to that of S ǫ + . Furthermore, we may find a shortneck Γ ǫ , such that Γ ǫ is a smooth hypersurface disjoint with S ǫ + , S ǫ − , and that S ǫ + ∪ Γ ǫ ∪ S ǫ − canbe glued together smoothly. Additionally let us require that on Γ ǫ each of the angular variables θ , . . . , θ d does not vary more than π/ . Also, the area of Γ ǫ is entailed to shrink to zero as ǫ → + . Finally, at the points e and (1 + η/ e , we glue to S ǫ + ∪ Γ ǫ ∪ S ǫ − the Euclidean half-planes Π + and Π − , respectively, such that Π ± are isomorphic copies R d + and that S ǫ := Π + ∪ S ǫ + ∪ Γ ǫ ∪ S ǫ − ∪ Π − (2.12)is a smooth hypersurface homeomorphic to R d ⊂ R d +1 .We conclude the construction by setting M ǫ := Ä S ǫ , ( ˜ ψ ǫ ) δ ij ä , where δ ij is the Euclidean metric on R d +1 . Note that for each ǫ > the manifold M ǫ ishomeomorphic to R d . The origin ∈ R d corresponds to the point that lies in the neck Γ ǫ forall ǫ > . One may think of each of the variables x i , i ∈ { , , . . . , d } in (2.8) and the ensuingarguments as the “time” variable analogous to x in Lemma 2.1, and view x = ( x , . . . , x d ) asbeing restricted to the diagonal. Thus, each angular variable θ ǫi behaves in the same way as θ ǫ in Lemma 2.1, and these variables are “synchronised”.In the above construction of M ǫ , we find that the Gauss map n ǫ of the immersion (in fact,embedding) ˜ ψ ǫ still takes the form of (2.8) with θ ǫi ( x ) ≡ ˜Θ ǫ ( x i ) . Moreover, each step of theconstruction can be performed with sufficiently small perturbations in any norm, say C ; thus k ψ ǫ − ˜ ψ ǫ k C ≤ Cη (2.13)for a universal constant C . So the second fundamental forms ˜II ǫ for ˜ ψ ǫ are also uniformly closeto II ǫ , say in the C -topology. In particular, as ˜II ǫ and II ǫ are both compactly supported on B (0 , , we deduce from (2.11) that (cid:12)(cid:12)(cid:12)(cid:12) k ˜II ǫ k L d ( R d ) − m · π (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cη, where C is a universal constant. Hence k ˜II ǫ k L d ( R d ) is uniformly bounded.Finally, let us consider the topological degree for ˜ ψ ǫ . By construction, Π + , Γ ǫ and Π − donot contribute to the degree. Also, the following holds independently of ǫ : deg Ä ˜ ψ ǫ |S ǫ ± ä = ± m . (2.14)Indeed, thanks to the definition of ψ ǫ and (2.13), the images under ˜ ψ ǫ of S ǫ ± are m times thenon-trivial generator of H d ( S d ) with the positive and negative orientations, respectively. Nowsuppose ¯ ψ were a limit { ˜ ψ ǫ } as an immersion of hypersurface, then (2.14) would have beenpreserved. Whereas, by (2.13)(2.9) and in light of the construction of the kernel K ǫ and the neck Γ ǫ , any pointwise subsequential limit of ˜ ψ ǫ has zero degree. So, in light of the diffeomorphism-invariant property of the topological degree, ¯ ψ cannot be an immersion modulo the action of Diff( R n ) . This completes the proof. (cid:3) Similar to the remarks ensuing the proof of Lemma 2.1, this counterexample can be lo-calised, and we can get a family of immersions of R d blowing up at an infinite discrete set. . Local Hölder Regularity
Consider the moduli space F ( δ, d ) := ® f ∈ W ,d ∩ C ∞ ( M ; R d +1 ) : f is an immersion , M is an d -dimensional hypersurface , M ∪ {∞} is smooth in S d +1 , k n k BMO( M ) ≤ δ, f ( M ) contains a fixed point ´ . (3.1)Heuristically, we show the following: if the Gauss maps of a family of smooth homeomorphic R d have uniformly small oscillations at all scales, then “a little” regularity persists in the limit.To state the result rigorously, we need the following Definition 3.1.
A set Ω ⊂ R d is a Hölder graph system if it can be locally represented by graphsof C ,γ -functions for some γ ∈ ]0 , . The notion of “graph systems” plays an essential role in the works [6, 1]. Note that wedo not require further geometric information for a Hölder graph system, e.g. , whether or not itrepresents a topological manifold or orbifold.Our main result of this section can now be stated as follows:
Theorem 3.2.
There exists a small constant δ > depending only on the dimension d , suchthat for any δ ∈ [0 , δ ] and any family of immersions { ψ ǫ } ⊂ F ( d, δ ) , we can find { σ ǫ } ⊂ Diff( R d ) such that, after passing to subsequences, { ψ ǫ ◦ σ ǫ } converges to a Hölder graph system. The proof is based on the framework and results developed by S. Semmes ([8, 9, 10]) onthe harmonic analysis on chord-arc surfaces with small constants:
Definition 3.3 (See Main Theorem, p.200 of [8]) . Let M be a hypersurface in R d +1 . It is achord-arc surface with small constant if γ > is small, or equivalently, η > is small. γ isdefined to be the smallest number such that ( γ The
BM O -norm of the Gauss map n is no larger than γ ; ( γ For each x ∈ M , R > and y ∈ B ( x, R ) , there holds | ( x − y ) · n x,R | ≤ γR .Here and hereafter f x,R := − R B ( x,R ) f = Vol − [ B ( x, R )] R B ( x,R ) f for each function f . On the otherhand, η > is the smallest number such that ( η For every x ∈ M , R > , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Vol Ä M ∩ B ( x, R ) ä Vol Ä B (0 , ä R d − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ η ;( η For any x, y ∈ M , d ( x, y ) ≤ (1 + η ) | x − y | . Here d denotes the geodesic distance on M . In fact, in the Main Theorem, p.200 of [8], the equivalence of the two conditions in Defini-tion 3.3 is proved, together with yet another two equivalent conditions ( α ) and ( β ) defined viaClifford–Cauchy integrals and Hardy spaces. Each of the numbers α, β, γ, η can be called “thechord-arc constant”.Our proof of Theorem 3.2 relies crucially on three results in [8, 9, 10]. Let us first discussthese results instinctively and non-rigorously by emphasising their geometric meanings. Theexact statements, quotations and explicit estimates will be presented in the proof. For M in the moduli space F ( δ, d ) , if δ is sufficiently small, then it is proved in [9, 10] that M is a chord-arc surface with small constant. In other words, when restricted F ( δ, d ) , ( γ implies ( γ . In particular, for each x ∈ M and R > , B ( x, R ) ∩ M stays close tothe hyperplane through x normal to the averaged Gauss map n x,R . • A chord-arc surface M with small constant can be “smoothed” in a small neighbourhoodaround any x ∈ M : there exists another chord-arc surface which is a Lipschitz graph,and which stays very close to M . • A chord-arc surface M with small constant has a “bi-Hölder” parametrisation by R d . One more issue before presenting the proof: we need
Definition 3.4.
Let ( X, d ) be a metric space. N ⊂ X is said to be a R -net if X = S z ∈N B ( z, R ) .Moreover, ˜ N is said to be a ˜ R -subnet of N if X = S ˜ z ∈ ˜ N B (˜ z, ˜ R ) , and if for each ˜ z ∈ ˜ N one canfind some z ∈ N such that B (˜ z, ˜ R ) ⊂ B ( z, R ) . In the above B ( • , • ) are the metric balls, and by an abuse of notations we also refer to { B ( z, R ) : z ∈ N } as the R -net. Proof of Theorem 3.2.
Assume
M ∈ F ( δ, d ) with δ ≤ δ to be chosen. Fix any t > , e.g. t = 10 − . By § , [9] one can find another chord-arc surface M t with the constant µ to bespecified, such that ≤ δ ≤ δ ≤ C ( d ) δ < µ. Next, in view of Eq. (3.7) and Lemma 3.8 in [9], M t ∩ B Ä x, (2 − + 10 − ) t ä is a Lipschitz graph with constant ≤ C µ for each x ∈ M , provided that µ = µ ( t, δ ) is chosenlarge enough. Here C = C ( d, δ ) . Under the same condition, M t can be taken sufficiently closeto M . More precisely, by Lemma 3.8 in [9], one may take dist( M t , M ) ≤ − t. Moreover, by Theorem 4.1 in [9], there exists a homeomorphism τ : M → M t such that max n k τ k C ,γ ( B ( x, t ) ∩M ) , k τ − k C ,γ ( B ( x, t )) ∩M t o ≤ C for all x ∈ M , (3.2)where C = C ( d, δ , t ) and the Hölder index is given by γ ≡ − C dδ (3.3)for a dimensional constant C (denoted by k in [9]). In fact, putting together Eqs. (1.3)(4.6) ,Lemma 5.5 in [9] and that ≤ δ ≤ δ , we may explicitly select C = C C δ ® (100 t ) C δ − · d δ ´ . (3.4)Here C = C ( d ) is a dimensional constant. Notice that our estimates (3.4)(3.2) are uniform in δ . We restrict to δ < ( C d ) − to ensure that γ > in (3.3).With the above explicit estimates at hand, we are ready to conclude the theorem. Byconsidering a compact exhaustion {M k } ր M , one may take M to be a bounded domain in It remains an open question if we change “bi-Hölder” to “bi-Lipschitz”; see [11] by T. Toro for discussions. Eq. (4,6) in [9] contains an index p ; for our purpose we can take it to be ( C γ ) − , by Theorem 4.1 in [9] d . Then we can take a (50 t ) -net N of M , whose cardinality is H ( N ) = C t − d for some geometric constant C = C ( d, γ ) ≡ C ( d, δ ) . Restricted to each member in the N ,the hypersurface M is C ,γ -parametrised by M t , a Lipschitz graph over (2 − + 10 − ) -balls.Using the quantitative estimates in the preceding paragraphs, we can refine N to a subnet ˜ N with cardinality C t − d , where C = C ( d, δ ) again, such that in each B ∈ ˜ N the set B ∩ M isparametrised by a C ,γ -homeomorphism with the Hölder norm bounded by C := C µ · C .To complete the proof, let us choose µ = 10 C ( d ) δ . By carefully tracing the dependence of the constants C , . . . , C in the above arguments, oneconcludes that C = C ( d, δ , t ) . But t = 10 − is fixed from the beginning of the proof, so C depends only on the dimension d and δ , the upper bound for the BM O -norm of the Gauss map.Therefore, the assertion now follows from the Arzela–Ascoli theorem, i.e. , the compactness of C ,γ ֒ → C ,γ ′ for γ ′ ∈ ]0 , γ [ . (cid:3) Three Further Questions Let the moduli space F ( A, E, p ) be as in § . Is the subspace F isom ( A, E, p ) := n ψ ∈ F ( A, E, p ) : ψ is an isometric immersion of a fixed manifold M o compact in its natural topology? For the end-point case p = 2 = d the answer is affirmative, incontrast to the unconstrained case for F ( A, E, p ) . The authors of [3] proved this via establishingthe weak continuity of the Gauss–Codazzi equations (the PDE system for the isometric immer-sion), with the help of a div-curl type lemma due to Conti–Dolzmann–Müller in [4]. What abouthigher dimensions d ≥ (and co-dimensions greater than )? That is, for a family of isometricimmersions of some fixed d -dimensional manifold with uniformly bounded second fundamentalforms in L d , is the subsequential limit an isometric immersion? Theorem 3.2 leaves open the possibility that the limiting objects of W ,d -bounded im-mersed hypersurfaces may be very irregular ( e.g. , the nowhere differentiable Weierstrass functionis C ,γ , or other fractals), even if the geometrical condition that the Gauss map is slowly oscillat-ing is enforced. Can we find natural geometrical conditions on the moduli space of d -dimensionalhypersurfaces with uniformly bounded second fundamental forms in L d , which is sufficient toensure higher regularities for the subsequential limits, e.g. , BV or Lipschitz? This is relatedto the problem of finding good parametrisations of chord-arc surfaces; see the discussions by S.Semmes [9] and T. Toro [11]. Theorem 2.2 shows that space of smooth hypersurfaces in R d +1 with uniformly L d -bounded second fundamental forms is non-compact modulo diffeomorphisms. Under what addi-tional conditions can we retain compactness? For the simplest case, under what extra geometricalor analytical assumptions is the space of topological S immersed in R with k II k L ≤ E compact? Acknowledgement . This work has been done during the author’s stay as a CRM–ISM post-doctoral fellow at Centre de Recherches Mathématiques, Université de Montréal and Institut desSciences Mathématiques. Siran Li would like to thank these institutions for their hospitality. The uthor is also indebted to Prof. Gui-Qiang G. Chen and Prof. Pengfei Guan for their continuoussupport and many insightful discussions on isometric immersions. References [1] P. Breuning, Immersions with bounded second fundamental form,
J. Geom. Anal. (2015), 1344–1386.[2] J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. (1970), 61–74.[3] G.-Q. Chen, S. Li, Global Weak Rigidity of the Gauss–Codazzi–Ricci Equations and Isometric Immersionsof Riemannian Manifolds with Lower Regularity, J. Geom. Anal. (2018), 1957–2007.[4] S. Conti, G. Dolzmann and S. Müller, The div-curl lemma for sequences whose divergence and curl arecompact in W − , , C. R. Acad. Sci. Paris Ser. I (2011), 175–178.[5] K. Corlette, Immersions with bounded curvature,
Geom. Dedicata (1990), 153–161.[6] J. Langer, A compactness theorem for surfaces with L p -bounded second fundamental form, Math. Ann. (1985), 223–234.[7] T. Rivière, Weak immersions of surfaces with L -bounded second fundamental form, in: Geometric Analysis,IAS/Park City Math. Ser. , vol. 22, pp.303–384, Amer. Math. Soc., Providence, 2016.[8] S. Semmes, Chord-arc surfaces with small constant. I,
Adv. Math. (1991), 198–223.[9] S. Semmes, Chord-arc surfaces with small constant. II: good parametrizations, Adv. Math. (1991), 170–199.[10] S. Semmes, Hypersurfaces in R n whose unit normal has small BMO norm,
Proc. Amer. Math. Soc. (1991), 403–412.[11] T. Toro, Surfaces with generalized second fundamental form in L are Lipschitz manifolds, J. Diff. Geom. (1994), 65–101. Siran Li: Department of Mathematics, Rice University, MS 136 P.O. Box 1892, Houston,Texas, 77251-1892, USA • Department of Mathematics, McGill University, Burnside Hall, 805Sherbrooke Street West, Montreal, Quebec, H3A 0B9, Canada