Learn more about Institutional subscriptions, Barbosa, J.L., do Carmo, M.: On the size of a stable minimal surface inR On the basis of this inequality, we obtain sufficient conditions for the existence and non-existence of a MOTS (along with outer trapped surfaces) in the domain, and for the existence of a minimal surface in its Jang graph, expressed in terms of various quasi-local mass quantities and the boundary geometry of the domain. The Zero-Moment Point (ZMP) [1] criterion, namely that outermost minimal surface is a minimal surface which is not contained entirely inside another minimal surface. The operators A - aK are intimately connected with the stability of minimal surfaces, the case a = 2 for surfaces in R3, and the case Q = 1 for surfaces in scalar flat 3-manifolds (see Theorem 4). Immediate online access to all issues from 2019. The UConn Summer School in Minimal Surfaces, Flows, and Relativity is a focused one-week program for graduate students and recent PhDs in geometric analysis, from 16th to 20th, July 2018. Barbosa, J.L., do Carmo, M.: A proof of the general isoperimetric inequality for surfaces. inequality to higher codimension, to non local perimeters and to non euclidean settings such as the Gauss space. Math. Stable approximations of a minimal surface problem with variational inequalities Nashed, M. Zuhair; Scherzer, Otmar; Abstract. 1-forms in the stability inequality with even slower decay towards the ends of the minimal surface than those considered previously. Arch. Destination page number Search scope Search Text Search scope Search Text Nashed, M.Zuhair; Scherzer, Otmar. [SSY], [CS] and [SS]. Jury. Finally, Section 6 gives an account on how the techniques developed for the isoperimetric inequality have been successfully applied to study the stability of other related inequalities. Therefore, the stability inequality (4) can be written in the form (5) 0 ≤ 2 Kf2 +4 f2 + |∇f|2, for any compactly supported function f on F. As noted in Section 2, the first eigenvalue of a complete minimal surface in hyperbolic space is bounded below by 1 4 the second variation of the area functional is non-negative. Tax calculation will be finalised during checkout. We identify a strong stability condition on minimal submanifolds that generalizes the above scenario. The Bernstein theorem was generalized by R. Osserman [lo] who showed that the statement is true Scand.5, 15–20 (1957), Polya, G., Szegö, G.: Isoperimetric inequalities of Mathematical Physics. Stability inequality : Carlo Schmid [CM11] Ch.1 §4 and Ch.1 §5 until Lemma 1.19 included : S.02.a: Bernstein problem : Daniel Paunovic [CM11] conclusion of Ch.1 §5 : S.02.b: 28.10. In particular, we consider the space of so-called stable minimal surfaces. A minimal surface S of general type and of maximal Albanese dimension satisfies the Severi inequality K 2 S ≥ 4χ (K S) (). 2 Let M be a minimal surface in the simply-connected space form of constant curvature a, and let D be a simply-connected compact domain with piecewise smooth boundary on M. Let A denote the second fundamental form of M . ... A theorem of Hopf and the Cauchy-Riemann inequality. Barbosa J.L., Carmo M.. (2012) Stability of Minimal Surfaces and Eigenvalues of the Laplacian. Stability inequality : Carlo Schmid [CM11] Ch.1 §4 and Ch.1 §5 until Lemma 1.19 included : S.02.a: Bernstein problem : Daniel Paunovic [CM11] conclusion of Ch.1 §5 : S.02.b: 28.10. For the systems that concern us in subsequent chapters, this area property is irrelevant. Stable approximations of a minimal surface problem with variational inequalities. https://doi.org/10.1007/BF01215521, Over 10 million scientific documents at your fingertips, Not logged in The Gauss-Bonnet Theorem (see Singer and Thorpe’s book). Circ. I.M.P.A., Rio de Janeiro: Instituto de Matematica Pura e Applicada 1973, Lichtenstein, L.: Beiträge zur Theorie der linearen partiellen Differentialgleichungen zweiter Ordnung von elliptischem typus. For the … Marginally trapped surfaces are of central importance in general relativity, where they play the role of apparent horizons, or quasilocal black hole bound-aries. PubMed Google Scholar, Barbosa, J.L., do Carmo, M. Stability of minimal surfaces and eigenvalues of the laplacian. These are minimal surfaces which, loosely speaking, are area-minimizing. 98, 515–528 (1976) Google Scholar. J. § are C1, parameterized by arclength, such that the tangent vector t = c0 is absolutely continuous. Minimal surfaces and harmonic functions : Fabian Jin [Oss86] §4 until Lemma 4.2 included : S.01.b: 15.10. Barbosa, J. L. (et al.) In §5 we prove a theorem on the stability of a minimal surface in R4, which does not have an analogue for 3-dimensional spaces. 3 Stable minimal surfaces and the first eigenvalue The purpose of this section is to obtain upper bounds for the first eigenvalue of stable minimal surfaces, which are defined as follows. Springer, Berlin, Heidelberg. Destination page number Search scope Search Text Search scope Search Text Index, vision number and stability of complete minimal surfaces. Processing of Telemetry Data Generated By Sensors Moving in a Varying Field (M113) D.J. (to appear), Bandle, C.: Konstruktion isomperimetrischer Ungleichungen der Mathematischen Physik aus solchen der Geometrie. 3 The Stability Estimate In this section we prove an estimate on the integral of the curvature which will be used in the proof of Bernstein’s theorem. In particular, F(E) F(K) = njKj whenever jEj= jKj. Preprint, Chern, S.S.: Minimal submanifolds in a Riemannian manifold. This inequality … If the free-surface flow of ice is defined as a variational inequality, the constraint imposed on the free surface by the bedrock topography is incorporated directly, thus sparing the need for ad hoc post-processing of the free boundary to enforce non-negativity of … a time symmetric Cauchy surface, then θ+ = 0 if and only if Σ is minimal. Stable minimal surfaces have many important properties. His key idea was to apply the stability inequality[See §1.2] to different well chosen functions. /Filter /FlateDecode $\endgroup$ – User4966 Nov 21 '14 at 7:12 68 0 obj The inequality was used by Simon in [Si] to show, among other things, that stable minimal hypercones of R n + 1 must be planar for n ≤ 6 and it was subsequently used to infer curvature estimates for stable minimal hypersurfaces, generalizing the classical work of Heinz [He], cf. 162, … So we get the minimal surface equation (MSE): div(ru p 1 + jruj2) We call the solution to this equation is minimal surface. >> 1 In [16] the expected inequality for area and charge has been proved for stable minimal surfaces on time symmetric initial data. We link these stability properties with the surface gravity of the horizon and/or to the existence of minimal sections. Ann. References Mini-courses will be given by. Lemma. Then, the stability inequality reads as R D jr˘j2 +2K˘2 >0. The Wul inequality states that, for any set of nite perimeter EˆRn, one has F(E) njKj1n jEj n 1 n; (1.1) see e.g. Then A 4πQ2 , (43) where A is the area of S and Q is its charge. Z.144, 169–174 (1975), Departamento de Matematica, Universidade Federal do Ceará, Fortaleza Ceará, Brasil, Instituto de Matematica Pura e Aplicada, Rua Luiz de Camões 68, 20060, Rio de Janeiro, R.J., Brasil, You can also search for this author in Then, take f = 1 in the stability inequality Q (f) 0 to nd jIIj2 + Ric g( ; ) d 0: Because jIIj2 0 and Ric g( ; ) >0 by assumption, this is a contradiction. Many papers have been devoted to investigating stability. Rational Mech. %PDF-1.5 Since minimal graphs are area-minimizing , it is natural to consider stable mini-mal hypersurfaces in Rn+1. Recall that if X is a minimal surface of general type over k, and ω X is the canonical bundle of X, then the Noether inequality asserts that h 0 (ω X) ⩽ 1 2 … © 2021 Springer Nature Switzerland AG. Mathematische Zeitschrift Amer. In [10] do Carmo and Peng gave J. https://doi.org/10.1007/978-3-642-25588-5_15. at the pointwise estimate. In this paper we develop a new approach for the stable approximation of a minimal surface problem associated with a relaxed Dirichlet problem in the space BV ... establish convergence and stability of approximate regularized solutions which are solutions of a family of variational inequalities. More precisely, a minimal surface is stable if there are no directions which can decrease the area; thus, it is a critical point with Morse index zero. Z.162, 245–261 (1978), Barbosa, J.L., do Carmo, M.: A necessary condition for a metric inR Pogorelov [22]). Definition 2. The Sobolev inequality (see Chapter 3). In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. volume 173, pages13–28(1980)Cite this article. Palermo33 201–211 (1912), Nitsche, J.: A new uniqueness theorem for minimal surfaces. stream Stability of surface contacts for humanoid robots: ... issue, as its dimension is minimal (six). Using the inequality of the Lemma for m = 2, we can improve the stability theorem of Barbosa and do Carmo [2]. 3 The key underlying property of the local versions of the inequality is the notion of stability, both for minimal hypersurfaces and for … The conjectured Penrose inequality, proved in the Riemannian case by 3. • When S is a K3 surface, Bayer … On the Size of a Stable Minimal Surface in R 3 Pages 115-128. ... J. Choe, The isoperimetric inequality for a minimal surface with radially connected boundary, MSRI preprint. It is obvious that a complete stable minimal hypersurface in \(\mathbb{H}^{n+1}(-1)\) has index 0. (i) The maximal quotients of the helicoid and the Scherk's surfaces … Exercise 6. It is well known that do Carmo and Peng [10], in 1979, proved that a complete stable minimal surface in R3 must be a plane (cf. If is a stable minimal … TheDirichlet problem forthe minimal surface problem istofindafunction u of minimal area A(u), as defined in (6) – (7), in the class BV(Ω) with prescribeddataφon∂Ω. Pure Appl. Guisti [3] found nonlinear entire minimal graphs in Rn+1. of Math.88, 62–105 (1968), Schiffman, M.: The Plateau problem for non-relative minima. For basics of hypersurface geometry and the derivation of the stability inequality, Simons’ identity and the Sobolev inequality on minimal hypersurfaces, [S] is an excellent reference. Subscription will auto renew annually. Math.-Verein.51, 219–257 (1941), Chen, C.C. We will usually assume that our curves c: (a;b)! Acad. The minimal surface equation 4/3 Calibrations 4/5 First variation and flux 4/8 Monotonicity 4/10 Extended Monotonicity 4/12 Bernstein's theorem 4/15 Stability 4/17 Stability continued 4/19 Stability stability stability 4/22 Bernstein theorem version 2 4/24 Weierstrass representation 4/26: Twistors 4/29 Theorem 3. the link-to-surface distance) while a fixed contact constraints all six DOFs of the end-effector link. [17, 15]. Rational Mech. << This is no longer true for higher codimensional minimal graphs in view of an example of Lawson and Osserman. We note that the construction of the index in this space (in the sense of Fischer-Colbrie [FC85]) in Section 4 is somewhat subtle. The slope inequality asserts that ω2 f ≥ 4g −4 g deg(f∗ωf) for a relative minimal fibration of genus g ≥ 2. The The Sobolev inequality (see Chapter 3). If (M;g) has positive Ricci curvature, then cannot be stable. A theorem of Micallef, which makes use of the complex stability inequality, states that any complete parabolic two-dimensional surface in four-dimensional Euclidean space is holomorphic }z"���9Qr~��3M���-���ٛo>���O���� y6���ӻ�_.�>�3��l!˳p�����W�E�7=���n���H��k��|��9s���瀠Rj~��Ƿ?�`�{/"�BgLh=[(˴�h�źlK؛��A��|{"���ƛr�훰bWweKë� �h���Uq"-�ŗm���z��'\W���܅��-�y@�v�ݖ�g��(��K��O�7D:B�,@�:����zG�vYl����}s{�3�B���݊��l� �)7EW�VQ������îm��]y��������Wz:xLp���EV����+|Z@#�_ʦ������G\��8s��H���� C�V���뀯�ՉC�I9_��):حK�~U5mGC��)O�|Y���~S'�̻�s�=�֢I�S��S����R��D�eƸ�=� ��8�H8�Sx0>�`�:Y��Y0� ��ժDE��["m��x�V� ... 1-forms in the stability inequality with even slower decay towards the ends of the minimal surface than those considered previously. The case involving both charge and angular momentum has been proved recently in [25]. Rend. In the context of multi-contact planning, it was advocated as a generalization ... do not mention how to compute the inequality constraints applying to these new variables. Indeed, the role of … A classi ca-tion theorem for complete stable minimal surfaces in three-dimensional Riemannian manifolds of nonnegative scalar curvature has been obtained by Fischer-Colbrie and Schoen [3]. n+1 to be isometrically and minimally immersed inM These are straightforward generalizations of Chen-Fraser-Pang and Carlotto-Franz results for free boundary minimal surfaces, respectively. [F] D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three manifolds, Invent. : Complete minimal surfaces with total curvature −2π. Proof. Let S be a stable minimal surface. Minimal surfaces of small total curvature : Martina Jorgensen We also obtain sharp upper bound estimates for the first eigenvalue of the super stability operator in the case of M is a surface in H 4. Math. Math. The proof of Theorem 1.2 uses crucially the fact that for two-dimensional minimal surfaces the sum of the squares of the principal curvatures 2 1 + 2 2 equals 2 1 2 = 2K, where Kis the Gauˇ curvature |since on a minimal surface 1 + 2 = 0. Remarks. Z. A Stability Inequality For a Class of Nonlinear Feedback Systems (M114) A.G. Dewey. We do not know the smallest value of a for which A-aK has a positive solution. Again, there is a chosen end of M3, and “contained entirely inside” is defined with respect to this end. Anal.45, 194–221 (1972), Lawson, Jr., B.: Lectures on Minimal Submanifolds. of Math.40, 834–854 (1939), Smale, S.: On the Morse index theorem. Comment. Mech.14, 1049–1056 (1965), Spruck, J.: Remarks on the stability of minimal submanifolds ofR A minimal surface is called stable if (and only if) the second variation of the area functional is nonnegative for all compactly supported deformations. The isoperimetric inequality for minimal surfaces (see, e.g., Chakerian, Proceedings of the AMS, volume 69, 1978). In: Tenenblat K. (eds) Manfredo P. do Carmo – Selected Papers. Rational Mech. Interestingly, it follows from a stability argument [43] that outermost minimal … Stable Approximations of a Minimal Surface Problem with Variational Inequalities M. Zuhair Nashed 1 and Otmar Scherzer 2 1 Department of Mathematical Sciences, University of … This is a preview of subscription content, access via your institution. Otis Chodosh (Princeton University) Geometric features of the Allen-Cahn equation; Ailana Fraser (University of British Columbia) Minimal surface methods in geometry Arch. We can run the whole minimal model program for the moduli space of Gieseker stable sheaves on P2 via wall crossing in the space of stability conditions. minimal surface. Brasil. Annals of Mathematics Studies21, Princeton: Princeton, University Press 1951, Simons, J.: Minimal Varieties in Riemannian manifolds. 2 [18] uses this notation for the intersection number mod 2 14 Proof. Math.10, 271–290 (1957), Osserman, R., Schiffer, M.: Doubly-connected minimal surfaces. If f: U R2!R is a solution of the minimal surface equation, then for all nonnegative Lipschitz functions : R3!R with support contained in U R, Z graph(f) jAj2 2d˙ C Z graph(f) jr graph(f) j 2d˙ We note that a noncompact minimal surface is said to be stable if its index is zero. Curves with weakly bounded curvature Let § be 2-manifold of class C2. Then!2 X=k 4˜(O X): Let us brie y introduce the history of this inequality. On the other hand, we can use either the Gauss–Bonnet theorem or the Jacobi equation to get the opposite bound. Pogorelov [22]). Math. It is well known that do Carmo and Peng [10], in 1979, proved that a complete stable minimal surface in R3 must be a plane (cf. In fact, all the known proofs are related to some sort of stability: geometric invariant theory in [CH] (in It is the curvature characteristic of minimal surfaces that is important. Math Z 173, 13–28 (1980). Abstract and Applied Analysis (1997) Volume: 2, Issue: 1-2, page 137-161; ISSN: 1085-3375; Access Full Article top Access to full text Full (PDF) How to cite top Jaigyoung Choe's main interest is in differential geometry. This notion of stability leads to an area inequality and a local splitting theorem for free boundary stable MOTS. Finally, we establish an index estimate and a diameter estimate for free boundary MOTS. In recent years, a lot of attention has been given to the stability of the isoperimetric/Wul inequality. $\begingroup$ The problem asks for the stability of the minimal surface. Since minimal graphs are area-minimizing , it is natural to consider stable mini-mal hypersurfaces in Rn+1. minimal surfaces: Corollary 2. strict stability of , we prove that a neighborhood of it in Mis iso- ... of a stable minimal surface ˆMwas in the proof of the positive mass theorem given by Schoen and Yau [17]. xڵ�r۸�=_�>U����:���N�u'��&3�}�%��$zI�^�}��)��l'm������@>�����^�'��x��{�gB�\k9����L0���r���]�f?�8����7N�s��? minimal surface M is a plane (Corollary 4). A minimal surface S of general type and of maximal Albanese dimension satisfies the Severi inequality K 2 S ≥ 4χ(K S ) ( [16]). Ann. And we will study when the sharp isoperimetric inequality for the minimal surface follows from that of the two flat surfaces. /Length 3024 43o �����lʮ��OU�-@6�]U�hj[������2�M�uW�Ũ� ^�t��n�Au���|���x�#*P�,i����˘����. Let X be a smooth minimal surface of general type over kand of maximal Albanese dimension. The main goal of this article is to extend this result in several directions. Part of Springer Nature. 1See [CM1] [CM2] for further reference. Suppose that M is connected and has finite genus, and suppose that x : M —>T\?/L is a complete, stable minimal immersion. The stability inequality (where D is the covariant derivative with respect to the Riemannian metric h) ⑤Dα⑤ 2 … Math. On the size of a stable minimal surface in R 3. A theorem of Micallef, which makes use of the complex stability inequality, states that any complete parabolic two-dimensional surface in four-dimensional Euclidean space is holomorphic The earliest result of this type was due to S. Bernstein [2] who proved this in the case that M is the graph of a function (stability is automatic in this case). Comm. minimal surface in hyperbolic space satisfy the following relation (Gauss’ lemma): K = −1− |B|2 2. Ci. It became again as a conjecture in [Ca,Re]. ;�0,3�r˅+���,cJ�"MbF��b����B;�N�*����? Arch. Stability of Minimal Surfaces and Eigenvalues of the Laplacian. If rankL = 1 or 2 then x(M) is a quotient of the plane, the helicoid or a Scherk's surface. - 85.214.85.191. 2. Math. Classify minimal surfaces in R3 whose Gauss map is one to one (see Theorem 9:4 in Osserman’s book). Barbosa and do Carmo showed [9] that an orientable minimal surface in R3 for which the area of the im-age of the Gauss map is less than 2 π is stable. Received 15 September 1979; First Online 01 February 2012; DOI https://doi.org/10.1007/978-3-642-25588-5_15 Gauss curvature for stable minimal surfaces in R3, which yielded the Bern-stein theorem for complete stable minimal surfaces in R3. For the integral estimates on jAj, follow the paper [SSY]. The Gauss-Bonnet Theorem (see Singer and Thorpe’s book). Publication: Abstract and Applied Analysis. Of course the minimal surface will not be stationary for arbitrary changes in the metric. Theorem 1.5 (Severi inequality). The stability inequality can be used to get upper bounds for the total curvature in terms of the area of a minimal surface. First, we prove the inequality for generic dynamical black holes. A stability criterion can be seen as a set of inequality constraints describing the conditions under which these equalities are preserved. (joint with R. Schoen) Mar 28, 2019 (Thur) 11:00-12:00 @ AB1 502a (Note special date and time.) Moreover, the minimal model is smooth. Nonlinear Sampled-Data Systems and Multidimensional Z-Transform (M112) A. Rault and E.I. A strong stability condition on minimal submanifolds Chung-Jun Tsai National Taiwan University Abstract: It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. Anal.58, 285–307 (1975), Peetre, J.: A generalization of Courant's nodal domain theorem. By plugging a … the inequality jSj 4pQ2 was proved for suitable surfaces. It is well-known that a minimal graph of codimension one is stable, i.e. Theorem 3.1 ([27, Theorem 0.2]). n. Math. Jber. Sakrison. For the minimal surface problem associated with (6) – (7), it is shown in Section 4 of Chapter … We establish the Noether inequality for projective 3-folds, and, specifically, we prove that the inequality vol (X) ≥ 4 3 p g (X) − 10 3 holds for all projective 3-folds X of general type with either p g (X) ≤ 4 or p g (X) ≥ 21, where p g (X) is the geometric genus and vol (X) is the canonical volume. Department of Mathematics Technical Report19, Lawrence, Kansas: University of Kansas 1968, Chern, S.S., Osserman, R.: Complete minimal surfaces in euclideann-space. Amer. Pages 441-456. A Reverse Isoperimetric Inequality and Extremal Theorems 3 1. n An. Classify minimal surfaces in R3 whose Gauss map is … A complex version of the stability inequality for minimal surfaces was derived, in-cluding curvature terms for the case of an underlying space which is not at. The minimal area property of minimal surfaces is characteristic only of a finite patch of the surface with prescribed boundary. %���� Minimal surfaces and harmonic functions : Fabian Jin [Oss86] §4 until Lemma 4.2 included : S.01.b: 15.10. At the same time, Fischer-Colbrie and Schoen [12], independently, showed Speaker: Chao Xia (Xiamen University) Title: Stability on … Assume that is stable. Helv.46, 182–213 (1971), Bol, G.: Isoperimetrische Ungleichungen für Bereiche und Flächen. Barbosa, João Lucas (et al.) In this paper we establish conditions on the length of the second fundamental form of a complete minimal submanifold M n in the hyperbolic space H n + m in order to show that M n is totally geodesic. uis minimal. Mat. Hence our theorem can be regarded as an extension of the results in [ 6 – 8 ]. J. Math.98, 515–528 (1976), Barbosa, J.L., do Carmo, M.: A proof of the general isoperimetric inequality for surfaces. In chemical reactions involving a solid material, the surface area to volume ratio is an important factor for the reactivity, that is, the rate at which the chemical reaction will proceed. J. Analyse Math.19, 15–34 (1967), Kaul, H.: Isoperimetrische Ungleichung und Gauss-Bonnet-Formel fürH-Flächen in Riemannschen Mannigfaltigkeiten. Anal.52, 319–329 (1973), Morrey, Jr., C.B., Nirenberg, L.: On the analyticity of the solutions of linear elliptic systems of partial differential equations. The surface-area-to-volume ratio, also called the surface-to-volume ratio and variously denoted sa/vol or SA:V, is the amount of surface area per unit volume of an object or collection of objects. For an immersed minimal surface in R3, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. can get a stability-free proof of the slope inequality. Pages 167-182. Destination page number Search scope Search Text Search scope Search Text Deutsch. Math. so the stability inequality (4) can be written in the form (5) 0 ≤ 2 Z K f2 +4 Z f2 + Z |∇f|2, for any compactly supported function f on F. As noted in Section 2, the first eigenvalue of a complete minimal surface in hyperbolic space is bounded below by 1 4. A complex version of the stability inequality for minimal surfaces was derived, in-cluding curvature terms for the case of an underlying space which is not at. interval (δ, 1], where δ > 0, and the extrinsic curvature of the surface satisfies the inequality \K e \ > 3(1 - δ)2/2δ. To learn the Moser iteration technique, follow [GT]. It was Severi who stated it as a theorem in [Se], whose proof was not correct unfortunately.
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