Curvature pinching theorems for minimal submanifolds

Abstract

<p>This Ph.D. is composed of the following three parts.<br />In Chapter 1, we apply Ros&rsquo; ideas and a different technique totally real minimal submanifolds in C P n+p(c ) and minimal submanifolds in a sphere, re&shy;spectively. In section 1, we give some preparation and fundamental formulas. In section 2, we have improved Simons&rsquo; pinching constant (for codimension p&gt; 3) and Shen Yibing&rsquo;s pinching constant for minimal submanifolds in a sphere. In section 3, we have improved N.Eriji&rsquo;s Ricci curvature pinching con&shy;stant for odd dimensional minimal submanifolds in a sphere. In section 4, we study the sectional curvature pinching problem for minimal submanifolds in a sphere. In section 5, we have solved the scalar curvature pinching problem for totally real minimal submanifolds in a complex projective space. In sec&shy;tion&nbsp; 6, we have got the Ricci curvature pinching theorems for 3-dimensional totally real minimal submanifolds in a complex projective space. In chapter&nbsp; 2, we have studied CR-submani folds in a almost Hermitian manifolds. It is a well-known fact that invariant submanifolds in Kaehler manifolds are minimal. In section 1, we introduce the concept of&nbsp; L-Kaehler manifolds and generalize this famous result to L-Kaehler manifolds. In sec&shy;<br />tion 2, we study integrability of distributions&nbsp; Dand&nbsp; D xof CR-submanifolds in an almost Hermitian manifold. In section 3, we prove that: Any totally umbilical non-trivial CR-submanifolds in a nearly Kaehler manifold are to&shy; tally geodesic if dim(Z)x ) &gt;&nbsp; 1 . In section 4, we prove that: Let A be a nearly Kaehler manifold with&nbsp; Hg &gt; 0. Then&nbsp; Nadmits no mixed foliate non - trivial&nbsp; C R &mdash; submanifolds. In section 5, we study the cohomology of&nbsp; C R &mdash;submanifolds of an almost Kaehler manifold. In section&nbsp; 6, we have got a suf&shy;<br />ficient condition that a compact totally real minimal submanifold of Sasakian<br />manifold is stable or unstable. In chapter 3, in section 1 and 2, we study the stability of surfaces with constant mean curvature in 3-dimensional hyperbolical space&nbsp; H3(c)and min&shy;imal surfaces in&nbsp; n &mdash; dimensional hyperbolic space&nbsp; H n(c).In section 3 and 4,<br />we compute the Gauss curvature of Gaussian image of minimal surfaces and<br />get an intrinsic characterization of affine&nbsp; 2-spheres.</p>

<p>Ova doktorska teza sastavljena je iz sledeća tri dela. U glavi&nbsp; 1 , primenjujemo ideje Rosa i različite tehnike na totalno realne minimalne podmnogostrukosti u&nbsp; C P n+p(c)i minimalne podmnogostrukosti u sferi, respektivno. U sekciji&nbsp; 1 , dajemo određenu pripremu i fundamen&shy; talne formule. U sekciji 2, pobolj&scaron;avamo Simpsonovu konstantu uklje&scaron;tenja (za kodimenziju&nbsp; p&gt; 3) i Shen Yibingovu konstantu uklje&scaron;tenja za mini&shy; malne podmnostrukosti u sferi. U sekciji 3, pobolj&scaron;avamo N. Erijijevu Ricijevu konstantu uklje&scaron;tenja krivine za neparno dimenzionalne minimalne podmnogostrukosti u sferi. U sekciji 4, ispitujemo problem uklje&scaron;tenja sekcione krivine za minimalne podmnogostrukosti u sferi. U sekciji 5, re&scaron;avamo problem uklje&scaron;tenja skalarne krivine za totalno realne minimalne mno&shy;gostrukosti u kompleksnom projektivnom prostoru. U sekciji&nbsp; 6, dobili smo teoreme uklje&scaron;tenja Riccijeve krivine za trodimenzionalne totalno realne minimalne mnogostrukosti u kompleksnom projektivnom prostoru. U glavi 2, ispitujemo CR-podmnogostrukosti u skoro Hermitovskim mnogostrukostima. Poznata je činjenica da su invarijantne podm nogostru&shy; kosti u Kaehlerovim mnogostrukostima minimalne. U sekciji&nbsp; 1 , uvodimo koncept L-Kaehlerovih mnogostrukosti i uop&scaron;tavamo gore navedeni poznati rezultat na L-Ivaehlerove mnogostrukosti. U sekciji 2, ispitujemo integrabilnost distribucija&nbsp; D i&nbsp; D xod CR-mnogostrukosti u skoro Hermitovsku mnogostrukost. U sekciji 3, mi dokazujemo: Sve totalno umbilične ne tri&shy;vijalne&nbsp; C R -mnogostrukost i u skoro Kaehlerovoj mnogostrukosti su totalno geodezijske ako je d im (D &ldquo;L) &gt;&nbsp; 1 . U sekciji 4, dokazujemo: Neka. je&nbsp; Nskoro Kaehlerovo jezgro sa&nbsp; H b&gt; 0 . Tada&nbsp; N ne dopu&scaron;ta nikakve me&scaron;ovito razlistane netrivijalne CR-podmnogostrukosti. U sekciji 5, ispitujemo kohomologiju&nbsp; C R-podmnogostrukosti skoro Kaehlerovih mnogostrukosti. U sekciji&nbsp; 6, dobijamo dovoljan uslov da kompaktna totalno realna minimalna mnogostrukost Sasakijanove mnogostrukosti bude stabilna ili nestabilna. U glavi 3, u sekcijama&nbsp; 1 i 2, ispitujemo stabilnost povr&scaron;i sa konstantnom srednjom krivinom u 3-dimenzionalnom hiperboločkom prostoru R 3 (c) i minimalne povr&scaron;i u n-dimenzionalnom hiperboličkom prostoru&nbsp; H n(c). U sekcijama 3 i 4, izračunavamo Gausovu krivinu Gausove slike minimalne povr&scaron;i i dobijamo unutra&scaron;nju karakterizariju afine&nbsp; 2-sfere.</p>