Tính hyperbolic đầy của miền hartogs

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❑❤❛♠❜❛② P❍❆❱■❙❆❨ ❚➼♥❤ ❤②♣❡r❜♦❧✐❝ ✤➛② ❝õ❛ ♠✐➲♥ ❍❛rt♦❣s ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✺ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❑❤❛♠❜❛② P❍❆❱■❙❆❨ ❚➼♥❤ ❤②♣❡r❜♦❧✐❝ ✤➛② ❝õ❛ ♠✐➲♥ ❍❛rt♦❣s ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣÷í✐ ữợ r ✲ ✷✵✶✺ ▲í✐ ❝❛♠ ✤♦❛♥ ❇↔♥ ❧✉➟♥ ✈➠♥ ♥➔② sü ự tổ ữợ sỹ ữợ ❝õ❛ ❚❙✳ ❚r➛♥ ❍✉➺ ▼✐♥❤✱ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ tr✉♥❣ t❤ü❝✳ ▲✉➟♥ ✈➠♥ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❝ỉ♥❣ ❜è tr♦♥❣ ❜➜t ❝ù ❝ỉ♥❣ tr➻♥❤ ♥➔♦✳ ❍å❝ ✈✐➯♥ ❑❤❛♠ ❜❛② P❍❆❱■❙❆❨ ❳→❝ ♥❤➟♥ ❝õ❛ tr÷ð♥❣ ❦❤♦❛ ❚♦→♥ ữớ ữợ r ▼✐♥❤ ✐ ▼ö❝ ❧ö❝ ▲❮■ ◆➶■ ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳ ⑩♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳ ✣❛ t↕♣ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳ ❑❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳ ●✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺✳ ❑❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ❤②♣❡r❜♦❧✐❝ ✈➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻✳ ❑❤æ♥❣ ❣✐❛♥ ♣❤ù❝ t❛✉t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỏ ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỏ ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✹ ✺ ✺ ✼ ✽ ✾ ✾ ❈❤÷ì♥❣ ✷✳ ❚➼♥❤ ❤②♣❡r❜♦❧✐❝ ✤➛② ❝õ❛ ♠✐➲♥ ❍❛rt♦❣s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✐✐ ▲❮■ ◆➶■ ✣❺❯ ◆❣❤✐➯♥ ❝ù✉ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❜➔✐ t♦→♥ ❝ì ❜↔♥ ♥❤➜t ❝õ❛ ❣✐↔✐ t➼❝❤ ♣❤ù❝ ❤②♣❡r❜♦❧✐❝✳ ự õ ữủ t ữợ õ ✤ë ❦❤→❝ ♥❤❛✉✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ t➻♠ ❦✐➳♠ ♥❤ú♥❣ ✤➦❝ tr÷♥❣ ❝❤♦ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♣❤ù❝ tị② þ❀ ❦❤↔♦ s→t t➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♥❤ú♥❣ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤ù❝ ❝ư t❤➸❀ ù♥❣ ❞ư♥❣ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤ù❝ ✈➔♦ ♥❤ú♥❣ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ❝õ❛ ❤➻♥❤ ❤å❝ ♣❤ù❝ ✈➔ ❣✐↔✐ t➼❝❤ ♣❤ù❝ ✳✳✳ ❚r♦♥❣ ♥❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙②✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♥❤ú♥❣ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤ù❝ ❝ư t❤➸ ❝ơ♥❣ ♥❤÷ ✈✐➺❝ t➻♠ ❤✐➸✉ ♥❤ú♥❣ ợ ổ ự r tữớ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝✳ ▼✐➲♥ ❍❛rt♦❣s t❤✉ë❝ ✈➔♦ sè ♥❤ú♥❣ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤ù❝ ♥❤÷ ✈➟②✳ ❈❤♦ X ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♣❤ù❝✱ ϕ : X → [−∞, ∞) ❧➔ ❤➔♠ ♥û❛ ❧✐➯♥ tư❝ tr➯♥ tr➯♥ X ✳ ▼✐➲♥ ❍❛rt♦❣s Ωϕ(X) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ Ωϕ (X) = (z, w) ∈ X × C : |w| < e−ϕ(z) ❘➜t ♥❤✐➲✉ t➼♥❤ ❝❤➜t rts ữủ t r ữợ q ❝õ❛ ❣✐↔✐ t➼❝❤ ♣❤ù❝ ❤②♣❡r❜♦❧✐❝✳ ▲✉➟♥ ✈➠♥ ✧ ❚➼♥❤ ❤②♣❡r❜♦❧✐❝ ✤➛② ❝õ❛ ♠✐➲♥ ❍❛rt♦❣s ✧ ♥❤➡♠ tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠✐➲♥ ❍❛rt♦❣s Ωϕ(X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ❱➔ ❝❤➾ r❛ ♠ët sè ❧ỵ♣ ❤➔♠ ✤❛ ỏ ữợ tr X (X) r ✤➛②✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❞ü❛ tr➯♥ ❦➳t q✉↔ ❝õ❛ ❜➔✐ ❜→♦✧ ❝♦♠♣❧❡t❡ ❤②♣❡r❜♦❧✐❝✐t② ♦❢ ❍❛rt♦❣s ❞♦♠❛✐♥s ✧ ❝õ❛ t→❝ ❣✐↔ ❉✳❉✳ ❚❤❛✐ ✈➔ ◆✳◗✳ ❉✐➺✉✳ ▲✉➟♥ ✈➠♥ ỗ tr tr õ õ ❝❤÷ì♥❣ ♥ë✐ ❞✉♥❣✱ ♣❤➛♥ ❦➳t ❧✉➟♥ ✈➔ ❞❛♥❤ ♠ư❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❈❤÷ì♥❣ ✶✿ ❚r➻♥❤ ❜➔② tê♥❣ q✉❛♥ ✈➔ ❤➺ t❤è♥❣ ❧↕✐ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝➛♥ t❤✐➳t ❝❤♦ ❝❤÷ì♥❣ s❛✉✳ ✶ ❈❤÷ì♥❣ ✷✿ ▲➔ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✱ tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ✤➛② ❝õ❛ ♠✐➲♥ ❍❛rt♦❣s Ωϕ(X) ✈➔ ❝❤➾ r❛ ♠ët sè ❧ỵ♣ ❤➔♠ ✤❛ ỏ ữợ tr X s rts Ωϕ (X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ❈✉è✐ ❝ò♥❣ ❧➔ ♣❤➛♥ ❦➳t ❧✉➟♥ tr➻♥❤ ❜➔② tâ♠ t➢t ❝→❝ ❦➳t q✉↔ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❧✉➟♥ ✈➠♥✳ ❇↔♥ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ữợ sỹ ữợ t t r ❍✉➺ ▼✐♥❤✳ ❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❈ỉ sỹ ữợ t t q tr q tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❊♠ ①✐♥ ❝↔♠ ì♥ ♣❤á♥❣ ✣➔♦ t↕♦✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥✱ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❱✐➺♥ ❚♦→♥ ❤å❝ ✈➔ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ ❡♠ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❳✐♥ ❝↔♠ ì♥ ✤➳♥ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ❧ỵ♣ ❝❛♦ ❤å❝ t♦→♥ ❑✷✶ ✤➣ ❧✉æ♥ ✤ë♥❣ ✈✐➯♥✱ ❝❤✐❛ s➫ ❦❤â ❦❤➠♥ ✈➔ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❜➔② tä ỏ t ỡ s s tợ ỳ ữớ t tr ❣✐❛ ✤➻♥❤ ❝õ❛ ♠➻♥❤✱ ♥❤ú♥❣ ♥❣÷í✐ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ q✉❛♥ t➙♠ ❣✐ó♣ ✤ï tỉ✐ ✈➔ ❧✉ỉ♥ ♠♦♥❣ ♠ä✐ tỉ✐ t❤➔♥❤ ❝æ♥❣✳ ❇↔♥ ❧✉➟♥ ✈➠♥ ❝❤➢❝ ❝❤➢♥ s➩ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❦❤✐➳♠ ❦❤✉②➳t✱ ✈➻ ✈➟② ❡♠ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ ỳ õ õ ỵ t ổ ✈➔ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ✤➸ ❧✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✳✳✳✳✳✳t❤→♥❣ ✹ ♥➠♠ ✷✵✶✺ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ❑❤❛♠ ❜❛② P❍❆❱■❙❆❨ ✷ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶✳ ⑩♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ●✐↔ sû X ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ Cn ✈➔ f : X → C ❧➔ ♠ët ❤➔♠ sè✳ ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ♣❤ù❝ t↕✐ x0 ∈ X tỗ t t t : Cn → C s❛♦ ❝❤♦ lim|h|→0 |f (x0 + h) − f (x0 ) − λ(h)| = 0, |h| tr♦♥❣ ✤â h = (h1, , hn) ∈ Cn ✈➔ |h| = ( ni=1 |hi|2) ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾♥❤ ❤➻♥❤ t↕✐ x0 ∈ X ♥➳✉ f ❦❤↔ ✈✐ ♣❤ù❝ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ♥➔♦ ✤â ❝õ❛ x0 ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ X ♥➳✉ f ❝❤➾♥❤ ❤➻♥❤ t↕✐ ♠å✐ ✤✐➸♠ t❤✉ë❝ X ✳ ▼ët →♥❤ ①↕ f : X Cm õ t t ữợ f = (f1, , fm), tr♦♥❣ ✤â fi = πi ◦ f : X → C, i = 1, , m ❧➔ ❝→❝ ❤➔♠ tå❛ ✤ë✳ ❑❤✐ ✤â f ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ X ♥➳✉ fi ❝❤➾♥❤ ❤➻♥❤ tr➯♥ X ✈ỵ✐ ♠å✐ i = 1, , m ⑩♥❤ ①↕ f : X → f (X) ⊂ Cn ✤÷đ❝ ❣å✐ ❧➔ s♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ♥➳✉ f ❧➔ s♦♥❣ →♥❤✱ ❝❤➾♥❤ ❤➻♥❤ ✈➔ f −1 ❝ô♥❣ ❧➔ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤✳ ✸ ✶✳✷✳ ✣❛ t↕♣ ♣❤ù❝ ❛✮ ✣à♥❤ ♥❣❤➽❛ ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ tæ♣æ ❍❛✉s❞♦r❢❢✳ ✰ (U, ) ữủ ởt ỗ ♣❤÷ì♥❣ ❝õ❛ X ✱ tr♦♥❣ ✤â U ❧➔ t➟♣ ♠ð tr♦♥❣ X ✈➔ ϕ : U → Cn ❧➔ →♥❤ ①↕✱ ♥➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿ ✐✮ ϕ(U ) ❧➔ t➟♣ ♠ð tr♦♥❣ Cn ✐✐✮ ϕ : U (U ) ởt ỗ ổ A = {(Ui, i)}iI ỗ ữỡ X ữủ ởt t ỗ t ✭❛t❧❛s✮ ❝õ❛ X ♥➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥ ✐✮ {Ui}i∈I ❧➔ ♠ët ♣❤õ ♠ð ❝õ❛ X ✐✐✮ ❱ỵ✐ ♠å✐ Ui, Uj ♠➔ Ui ∩ Uj = ∅, →♥❤ ①↕ ❧➔ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤✳ ❳➨t ❤å ❝→❝ ❛t❧❛s tr➯♥ X ❍❛✐ ❛t❧❛s A1, A2 ✤÷đ❝ ❣å✐ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ♥➳✉ ❤đ♣ A1 ∪ A2 ❧➔ ♠ët ❛t❧❛s✳ ✣➙② ❧➔ ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ t➟♣ ❝→❝ ts ộ ợ tữỡ ữỡ ởt trú ❦❤↔ ✈✐ ♣❤ù❝ tr➯♥ X ✱ ✈➔ X ❝ị♥❣ ✈ỵ✐ ❝➜✉ tró❝ ❦❤↔ ✈✐ ♣❤ù❝ tr➯♥ ♥â ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤❛ t↕♣ ♣❤ù❝ n ❝❤✐➲✉✳ ϕj ◦ ϕi −1 : ϕi (Ui ∩ Uj ) → ϕj (Ui ∩ Uj ) ❜✮ ⑩♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❣✐ú❛ ❝→❝ ✤❛ t↕♣ ♣❤ù❝ ●✐↔ sû M, N ❧➔ ❝→❝ ✤❛ t↕♣ ♣❤ù❝✳ ⑩♥❤ ①↕ ❧✐➯♥ tö❝ f : M → N ữủ tr M ợ ỗ ữỡ (U, ) M ởt ỗ ữỡ (V, ) N s f (U ) ⊂ V t❤➻ →♥❤ ①↕ ψ ◦ f ◦ ϕ−1 : ϕ(U ) → ψ(V ) ✹ ❧➔ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤✳ ❍❛② t÷ì♥❣ ✤÷ì♥❣✱ ✈â✐ ♠å✐ x M, y N, tỗ t ỗ ữỡ (U, ) (V, ) t x ✈➔ y t÷ì♥❣ ù♥❣ s❛♦ ❝❤♦ ❧➔ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤✳ ●✐↔ sû f : M → N ❧➔ s♦♥❣ →♥❤ ❣✐ú❛ ❝→❝ ✤❛ t↕♣ ♣❤ù❝✳ ◆➳✉ f ✈➔ f −1 ❧➔ ❝→❝ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ t❤➻ f ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ s♦♥❣ ❝❤➾♥❤ ❤➻♥❤ ❣✐ú❛ M ✈➔ N ✳ ψ ◦ f ◦ ϕ−1 : ϕ(U ) → ψ(V ) ✶✳✸✳ ❑❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ●✐↔ sû Z ❧➔ ✤❛ t↕♣ ♣❤ù❝✳ ▼ët ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ✤â♥❣ ❧➔ ♠ët t➟♣ ❝♦♥ ✤â♥❣ ❝õ❛ Z ♠➔ ✈➲ ♠➦t ✤à❛ ♣❤÷ì♥❣ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❤ú✉ ❤↕♥ ữỡ tr t ự ợ x0 X tỗ t V x tr Z ✈➔ ❤ú✉ ❤↕♥ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ ϕ1, , ϕm tr➯♥ V s❛♦ ❝❤♦ X X ∩ V = {x ∈ V |ϕi (x) = 0, i = 1, , m} ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ♣❤ù❝ tr♦♥❣ ✤❛ t↕♣ ♣❤ù❝ Z ✳ ❍➔♠ f : X C ữủ ợ ộ x X tỗ t ởt U (x) ⊂ Z ✈➔ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ fˆ tr➯♥ U s❛♦ ❝❤♦ fˆ|U ∩X ⇒ f |U ∩X ●✐↔ sû f : X → Y ❧➔ →♥❤ ①↕ ❣✐ú❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ X ✈➔ Y ✳ f ữủ ợ ộ ❤➻♥❤ g tr➯♥ ♠ët t➟♣ ❝♦♥ ♠ð V ❝õ❛ Y ✱ ❤➔♠ ❤ñ♣ g ◦ f ❧➔ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ f −1(V ) ✶✳✹✳ ●✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ✣➦t Hol(X, Y ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ tø ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ X tợ ởt ổ ự Y ữủ tr tỉ♣ỉ ❝♦♠♣❛❝t ♠ð✳ ✺ ❜ð✐ ❱ỵ✐ < r < ∞, t❛ ✤➦t Dr = {z ∈ C : |z| < r}; D1 = D ❚r➯♥ ✤➽❛ ✤ì♥ ✈à ♠ð D, t❛ ①➨t ❦❤♦↔♥❣ ❝→❝❤ ❇❡r❣♠❛♥ ✲ P♦✐♥❝❛r➨ ❝❤♦ + |a| ; ∀a ∈ D − |a| z1 − z2 1+| | − z z2 , ∀ z1 , z2 ∈ D ρD (z1 , z2 ) = n |z1 − z2 | 1−| | − z z2 ρD (0, a) = n ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝✱ p, q ❧➔ tũ ỵ X t p0 = p, p1, , pk = q ❝õ❛ X ✱ ❞➣② ✤✐➸♠ a1, a2, , ak ❝õ❛ D ✈➔ ❞➣② ❝→❝ →♥❤ ①↕ f1, , fk tr♦♥❣ Hol(D, X) t❤ä❛ ♠➣♥ fi (0) = pi−1 , fi (ai ) = pi , ∀i = 1, , k ❚❛ ❣å✐ t➟♣ ❤ñ♣ α = {p0, , pk , a1, , ak , f1, , fk } ❧➔ ♠ët ❞➙② ❝❤✉②➲♥ ❝❤➾♥❤ ❤➻♥❤ ♥è✐ p ✈➔ q tr♦♥❣ X ✳ ❱ỵ✐ ♠é✐ ❞➙② ❝❤✉②➲♥ ♥❤÷ ✈➟②✱ t❛ ❧➟♣ tê♥❣ ki=1 pD (0, ai) ✣➦t dX (p, q) = infα ki=1 pD (0, ai), α ∈ Ωp,q , tr♦♥❣ ✤â Ωp,q ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❞➙② ❝❤✉②➲♥ ❝❤➾♥❤ ❤➻♥❤ ♥è✐ p ✈➔ q tr♦♥❣ X ✳ ❉➵ t❤➜② dX : X × X −→ R ❧➔ ♠ët ❣✐↔ ❦❤♦↔♥❣ ❝→❝❤ ✈➔ ❣å✐ ❧➔ ❣✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ X ✳ ❚❛ ❝â t❤➸ ❞➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✤➙② ❝õ❛ dX ✿ i) dX ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ii) ◆➳✉ f : X → Y ❧➔ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❣✐ú❛ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ t❤➻ f ❧➔♠ ❣✐↔♠ ❦❤♦↔♥❣ ❝→❝❤ ✤è✐ ✈ỵ✐ ❣✐↔ ❦❤♦↔♥❣ ❝→❝❤ ❑♦❜❛②❛s❤✐✱ ♥❣❤➽❛ ❧➔ dX (p, q) ✳ dY (f (p), f (q)), iii) dD = ρD ✳ ✻ ∀ p, q ∈ X ❤②♣❡r❜♦❧✐❝ ✤➛② ♥➯♥ X ❧➔ t❛✉t✱ ❜➡♥❣ ❝→❝❤ ❧➜② ❞➣② ❝♦♥ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ ❞➣② {fj } ❤ë✐ tö ✤➲✉ ✤à❛ ữỡ tr D tợ ởt f ∈ Hol(D, Ωϕ (X))✳ ❉➵ t❤➜② f (w) = (z0, rw)✳ ✣✐➲✉ ♥➔② ❝❤♦ t❛ r|w| < e−ϕ(z ), ∀ w ∈ D✱ ✈➔ ❞♦ ✤â r ≤ e−ϕ(z )✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✶✮✳ ❱➟② ϕ ❧➔ ❧✐➯♥ tư❝ tr➯♥ X ✳ ❈✉è✐ ❝ị♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ϕ ỏ ữợ tr X ỵ rss rs ữủ õ tr t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ¯ X)✳ ❚❛ ♠✐♥❤ r➡♥❣ ϕ g ỏ ữợ ợ g Hol(D, X) ∩ (D, ①➨t ♠✐➲♥ ❍❛rt♦❣s ♥❤÷ s❛✉✿ 0 Ωϕ◦g (X) = (z, w) : z ∈ D, |w| < e−(ϕ◦g) ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ Ωϕ◦g (D) ỗ sỷ ữủ s tỗ ¯ Ωϕ◦g (D)) t❤ä❛ ♠➣♥ ✿ t↕✐ ♠ët ❞➣② {ϕj } tr♦♥❣ Hol(D, Ωϕ◦g (X)) ∩ C(D, ¯ ⊂ Ωϕ◦g (D), ∀n ≥ 1✳ i) ϕn (D) ¯ ✤➳♥ ϕ∗ s❛♦ ❝❤♦ ϕ∗ (∂D) ⊂ Ωϕ◦g (D)✳ ii) {ϕn } ❤ë✐ tö ✤➲✉ tr➯♥ D iii) ϕ∗ (D) ⊂ Ωϕ◦g (D)✳ ✣➦t ψ(λ, w) = (g(λ), w), ∀(λ, w) ∈ D × C✳ ❳➨t ❞➣② ❝→❝ →♥❤ ①↕ {ϕ˜n} ①→❝ ✤à♥❤ ❜ð✐ ϕ˜n = ψ ◦ ϕn✳ ❚ø ✭✐✮ ✈➔ ✭✐✐✮ t❛ ❝â ˜n (∂D) n≥1 ϕ ∪ (ψ ◦ ϕ∗ )(∂D) ⊂⊂ Ωϕ (X) ❱➻ ✈➟②✱ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ ♠ët t➟♣ ❝♦♥ ♠ð ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ U ❝õ❛ Ωϕ (X) ❝❤ù❛ n≥1 ϕ˜n (∂D) ∪ (ψ ◦ )(D) õ tỗ t n0 ợ z0 ∈ D ✤õ ❣➛♥ ∂D s❛♦ ❝❤♦ ϕ˜n(z0) ∈ U ✈ỵ✐ ♠å✐ n ≥ n0✳ ❚❛ ✤➦t✿ F = {f ∈ Hol(D, Ωϕ (X)) : f (z0 ) ∈ U } ✶✹ ❱➻ Ωϕ(X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛② ♥➯♥ Ωϕ(X) ❧➔ t❛✉t✳ ❉♦ ✈➟② F ❧➔ ❝❤✉➞♥ t➢❝✳ ❱➻ ❦❤æ♥❣ ❝â ❞➣② ❝♦♥ tr♦♥❣ F ❝â t❤➸ ❧➔ ♣❤➙♥ ❦ý ❝♦♠♣❛❝t ✈➔ ϕ˜n ∈ F, ✈ỵ✐ ♠å✐ n ≥ n0 ♥➯♥ t❛ ❝â →♥❤ ①↕ ❣✐ỵ✐ ❤↕♥ ψ ◦ ϕ∗ t❤✉ë❝ F ✳ ✣➦❝ ❜✐➺t ψ ◦ ϕ∗ →♥❤ ①↕ D ✈➔♦ Ωϕ(X)✱ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✐✐✐✮✳ ❱➻ g (D) ỗ r g ỏ ữợ ỵ ữủ ự ♠✐♥❤✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ t✐➳♣ t❤❡♦✱ t❛ ♥❤➢❝ ❧↕✐ ▼➺♥❤ ✤➲ s❛✉✿ ▼➺♥❤ ✤➲ ✷✳✹✳✭❬✼❪✱ ♣✺✺✮✳ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝✱ a ∈ X ✈➔ ❝→❝ số ữỡ , õ tỗ t sè c > s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ δ > 0✱ ♠å✐ ❝➦♣ ✤✐➸♠ (p, q) t❤✉ë❝ ❤➻♥❤ ❝➛✉ ♠ð U (a, ρ) = {b ∈ X : dX (a, b) < ρ} ❝â t❤➸ ✤÷đ❝ ♥è✐ ❜ð✐ ♠ët ❞➙② ❝❤✉②➲♥ β ❝→❝ ✤➽❛ ❝❤➾♥❤ ❤➻♥❤ ❝â ✤ë ❞➔✐ l(β) < C(dX (p, q) + δ) ♥➡♠ tr♦♥❣ U (a; 3ρ + ε)✳ ✣➦t ❜✐➺t dU (a;3ρ+ε)(p, q) ≤ C.dX (p, q), ∀ p, q ∈ U (a; ρ)✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② < r < ❧➔ sè ❞÷ì♥❣ ①→❝ ✤à♥❤ ❜ð✐ dD (0, r) = ε✱ ✈➻ ✈➟② ✤➽❛ Dr = {z ∈ C, |z| < r} ❜➟❝ ❦➼♥❤ r ❝â ❜→♥ ❦➼♥❤ ε ù♥❣ ✈ỵ✐ ❦❤♦↔♥❣ ❝→❝❤ P♦✐♥❝❛r➨ dD ✳ ❚❛ ❝❤å♥ C t❤ä❛ ♠➣♥ dDr (0, x) ≤ C.dD (0, x) ✈ỵ✐ x ∈ D ✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ C t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ✱ t❛ ♥è✐ p, q ∈ U (a, ρ) ❜ð✐ ♠ët ❞➙② ❝❤✉②➲♥ α ❝→❝ ✤➽❛ ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣ X ❝â ✤ë ❞➔✐ l(α) < dX (p, q) + δ < 2ρ ✈ỵ✐ δ ✤õ ♥❤ä✳ ❱➻ ✤ë ❞➔✐ ❝õ❛ ❧✐➯♥ ❦➳t |α| ❜➨ ❤ì♥ 2ρ ♥➯♥ |α| ❜à ❝❤➦♥ tr♦♥❣ U (a, 3ρ)✳ ▲➜② fi : D → X ❧➔ ✤➽❛ ❝❤➾♥❤ ❤➻♥❤ t❤ù i ❝õ❛ ❞➙② ❝❤✉②➲♥ α ❜✐➳♥ ai, bi ∈ D t❤➔♥❤ pi−1, pi ∈ X ✳ ❑❤æ♥❣ ♠➜t tê♥❣ q✉→t ✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t = ✈➔ |bi| < 2r ❱➻ pi−1 ∈ U (a, 3ρ) ♥➯♥ fi(Dr ) ⊂ U (a, 3ρ + ε), ✈➻ ✈➟② ♥➳✉ z ∈ Dr , t❤➻ dD (0, z) < ε ✈➔ dX (pi−1 , fi (z)) = dX (fi (0), fi (z)) < ε ❇➡♥❣ ❝→❝❤ t❤✉ ❤➭♣ ✤➽❛ ❝❤➾♥❤ ❤➻♥❤ fi : D → X tr➯♥ Dr ✱ t❛ ❝â ❞➙② ❝❤✉②➲♥ β ❧➔ ❞➙② ❝❤✉②➲♥ ♥è✐ p ✈➔ q ♥➡♠ tr♦♥❣ U (a, 3ρ + ε) ✈➔ l(β) ≤ r C.l(α) < C.(dX (p, q) + ) tũ ỵ ♥➯♥ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❍➺ q✉↔ ✷✳✺✳ ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ✱ a X, > õ tỗ t sè C > s❛♦ ❝❤♦ ✈ỵ✐ p, q ∈ U (a, ρ) = {b ∈ X : dX (a, b) < ρ} t❛ ❝â dU (a,4ρ) (p, q) ≤ C.dX (p, q) ỵ sỷ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ❤②♣❡r❜♦❧✐❝✱ ϕ ❧➔ ♠ët ❤➔♠ ỏ ữợ tử tr tỹ tr X t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t s❛✉✿ ❱ỵ✐ ♠é✐ ✤✐➸♠ ❜✐➯♥ (z0 , w0 ) ❝õ❛ Ωϕ (X) ✈ỵ✐ z0 ∈ X tỗ t ởt V z0 tr♦♥❣ X ✈➔ ♠ët →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ f tø Ωϕ (X) ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ ❤②♣❡r❜♦❧✐❝ ✤➛② Y s❛♦ ❝❤♦ ❞➣② f (zn , wn ) ❦❤ỉ♥❣ ❧➔ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ tr♦♥❣ Y ✈ỵ✐ ❜➜t ❦ý ❞➣② {(zn , wn )} ❤ë✐ tư tỵ✐ (z, w)✳ ❑❤✐ ✤â Ωϕ (X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ❈❤ù♥❣ ♠✐♥❤ ✿ ❚❤❡♦ ❇ê ✤➲ ✷✳✾✱ Ωϕ(X) ❧➔ ❤②♣❡r❜♦❧✐❝✳ ❈❤ó♥❣ t❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ Ωϕ(X) r tt r tỗ t ởt ❞➣② ❈❛✉❝❤② {pk }k≥1 = {(zk , wk )}k≥1 tr♦♥❣ Ωϕ (X) s❛♦ ❝❤♦ {pk } ❦❤ỉ♥❣ ❤ë✐ tư tỵ✐ ❜➜t ❦ý ✤✐➸♠ ♥➔♦ tr♦♥❣ Ωϕ(X)✳ ❚❤❡♦ t➼♥❤ ❝❤➜t ❣✐↔♠ ❦❤♦↔♥❣ ❝→❝❤✱ {zk } ❧➔ ♠ët ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✳ ❱➻ X ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛② ♥➯♥ ❞➣② ♥➔② ❤ë✐ tư tỵ✐ z0 ∈ X ✳ ●✐↔ sû U ❧➔ ♠ët ❧➙♥ ❝➟♥ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ❝õ❛ z0 ∈ X ✳ ❇➡♥❣ ❝→❝❤ ❧➜② ♠ët ❞➣② ❝♦♥ ♥➳✉ ❝➛♥ t❤✐➳t ✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ {zk }k≥1 ⊂ U ✳ ❚ø ✤â s✉② r❛ {(zk , wk )}k≥1 ⊂ U × ∆ ⊂ U × C✱ tr♦♥❣ ✤â ∆ ❧➔ ✤➽❛ {w : |w| < e− inf ϕ(z) }✳ ❇➡♥❣ ❝→❝❤ ❧➜② ♠ët ❞➣② ❝♦♥ ♥➳✉ ❝➛♥ t❤✐➳t✱ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ ❞➣② {pk }k≥1 ❤ë✐ tư tỵ✐ ✤✐➸♠ tr♦♥❣ ∂(Ωϕ(X))✳ ❚ø ❣✐↔ t❤✐➳t✱ t❛ ❝â t❤➸ ❧➜② ♠ët ❧➙♥ ❝➟♥ V ❝õ❛ z0 tr♦♥❣ X ✱ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ f tr♦♥❣ π−1(V ) ✈➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❤②♣❡r❜♦❧✐❝ ✤➛② Y t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➣ ❝❤♦✳ ▲➜② < ρ < 51 inf{dX (z0, x) :∈ X\V }✱ ✈➔ N z∈U ✶✻ ✤õ ❧ỵ♥ s❛♦ ❝❤♦ pn ∈ U(p ,ρ) ✈ỵ✐ ♠å✐ n ≥ N ✳ ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ t➼♥❤ ❝❤➜t ❣✐↔♠ ❦❤♦↔♥❣ ❝→❝❤ t❛ ❝â N dX (z0 , zn ) ≤ p, ∀ n ≥ N ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ U (pN , 4ρ) ⊂ π−1(V )✳ ❚❤➟t ✈➟②✱ ♥➳✉ dΩ (X) (pN , ρ) < 4ρ t❤➻ ϕ dX (π(p), z0 ) ≤ dX (π(p), zN ) + dX (zN , z0 ) ≤ dΩϕ (X) (p, pN ) + ρ < 5ρ ≤ dX (z0 , X\V ) ❉♦ ✈➟② π(p) ∈ V ✱ t❛ ✤÷đ❝ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❍➺ q✉↔ ✷✳✺ t❛ ❝â C1, C2 > t❤ä❛ ♠➣♥✿ dπ−1 (V ) (pn , pN ) ≥ dU (pN ,4ρ) (pn , pN ) ≤ C1 dΩϕ (X) (pn , pN ) < C2 , ∀ n > N ✳ ▼➦t ❦❤→❝✱ t❛ ❝â dπ−1(V ) (pn , pN ) ≥ dY (f (pn ), f (pN )), ∀ n > N ❱➻ Y ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛② ✈➔ ❞➣② {f (pn)} ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ tr♦♥❣ Y, t❛ ❝â ♠➙✉ t❤✉➝♥✳ ❱➟② ✤à♥❤ ỵ ữủ ự ứ ỵ tr t õ t q s ỵ X ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ❤②♣❡r❜♦❧✐❝✱ ϕ ❧➔ ♠ët ❤➔♠ ✤❛ ỏ ữợ tử tr tỹ tr X t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t✿ ❱ỵ✐ ♠å✐ ✤✐➸♠ ❜✐➯♥ (z0 , w0 ) (X) z0 X, tỗ t↕✐ ♠ët ❧➙♥ ❝➟♥ V ❝õ❛ z0 tr♦♥❣ X, ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ f tr➯♥ Ωϕ (V ) s❛♦ ❝❤♦✿ |f (z, w)| < 1, ∀ (z, w) ∈ Ωϕ (V ) lim(z,w)→(z0 ,w0 ) |f (z, w)| = ❑❤✐ ✤â Ωϕ (X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ✶✼ ✣à♥❤ ♥❣❤➽❛ ✷✳✽✳ ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝✱ ϕ ❧➔ ❤➔♠ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ tr➯♥ X ✳ ❚❛ ♥â✐ r➡♥❣ ϕ ❝â t➼♥❤ ❝❤➜t (S) t↕✐ ♠ët ✤✐➸♠ a X tỗ t ởt Ua ❝õ❛ a tr♦♥❣ X ✈➔ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ fa tr➯♥ Ua t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ (i) ϕ(z) ≥ log |fa (z)|, ∀ z ∈ Ua ✱ (ii) ϕ(a) = log |fa (a)|✳ ◆❤➟♥ ①➨t✳ ●✐↔ sû ϕ ❧➔ tr➯♥ ❤➔♠ ♥û❛ ✤÷đ❝ ❧✐➯♥ tư❝ tr➯♥ X ✳ ●✐↔ t❤✐➳t r➡♥❣ ϕ ❝â t➼♥❤ ❝❤➜t (S) t↕✐ ♠å✐ ✤✐➸♠ ❝õ❛ X ✱ ❦❤✐ ✤â ❞➵ ❦✐➸♠ tr❛ ϕ ỏ ữợ tr X ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ❇ê ✤➲ ✷✳✾✳ ●✐↔ sû X ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✱ ϕ ❧➔ ♠ët ❤➔♠ ✤❛ ✤✐➲✉ ❤á❛ ữợ tử tr tỹ tr X t❤✐➳t r➡♥❣ ϕ ❝â t➼♥❤ ❝❤➜t (S) t↕✐ ♠å✐ ✤✐➸♠ ❝õ❛ X ✳ ❑❤✐ ✤â Ωϕ (X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ự ởt tũ ỵ (z0, w0) ∂(Ωϕ(X)) ✈ỵ✐ z0 ∈ X ✳ ❱➻ ϕ ❝â t➼♥❤ ❝❤➜t (S) t↕✐ z0 ♥➯♥ t❛ ❝â t❤➸ ❧➜② ♠ët ❧➙♥ ❝➟♥ ❜➨ U ❝õ❛ z0 ✈➔ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ fz t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✐✮ ✈➔ ✭✐✐✮ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ✷✳✽✳ ✣➦t✿ f (z, w) = wfz (z), ∀(z, w) ∈ π−1(U )✳ ❘ã r➔♥❣ f ❧➔ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ π−1(U )✳ ❚❛ ❝â✿ |f (z, w)| = |wfz (z)| ≤ |weϕ(z)| < 1, ∀(z, w) ∈ π−1(U ) ❉♦ |f (z0, w0)| = 1, t❤❡♦ ✣à♥❤ ỵ t õ ự ✷✳✶✵✳ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝✳ ▼ët ❤➔♠ ϕ : X → [−∞, ∞) ❣å✐ ❧➔ ✤❛ ✤✐➲✉ ỏ ữợ t tr X ợ ♥❤ó♥❣ ✤à❛ ♣❤÷ì♥❣ j : X → Ω ⊂ Cn, ϕ ❧➔ ❤↕♥ ❝❤➳ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ♠ët ❤➔♠ ✤❛ ỏ ữợ t tr ỵ s ✤➙② tr➻♥❤ ❜➔② ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ t➼♥❤ ❤②♣❡r❜♦❧✐❝ ✤➛② rts (X) 0 ỵ ✷✳✶✶✳ ❬✶✶❪ ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ●✐↔ t❤✐➳t r➡♥❣ ϕ ❧➔ ♠ët ❤➔♠ ✤❛ ✤✐➲✉ ỏ ữợ t tr X õ (X) ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ t↕✐ ♠é✐ ✤✐➸♠ a ∈ X ✱ ❤➔♠ ϕ ❝â t➼♥❤ ❝❤➜t (S)✳ ❱➻ ❜➔✐ t♦→♥ ❧➔ ✤à❛ ♣❤÷ì♥❣✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ q✉❛♥❤ ✤✐➸♠ a, X ❧➔ ♠ët t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ ❝õ❛ ♠ët t➟♣ ♠ð Y tr♦♥❣ Cn ỏ ữợ t tr X tỗ t ởt ỏ ữợ t tr Y s |X = ϕ✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ ˜ + pa (z) = 21 ϕ(a) n j,k=1 n j=1 ∂ ϕ˜ (a)(zj − aj )+ ∂zj ∂ ϕ˜ (a)(zj − aj )(zk − ak ) ∂zj ∂zk ❚ø ✤â t❛ ❝â ϕ(a) ˜ = Re(2pa (a))✳ ❚❛ ①➨t ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ❝õ❛ ϕ˜ t↕✐ a ϕ(z) ˜ = ϕ(a) ˜ + 2Re + 21 + n j,k=1 n j,k=1 ˜ n ∂ϕ (a)(zj j=1 ∂zj − aj ) ∂ ϕ˜ (a)(zj − aj )(zk − ak ) ∂zj , ∂zj ∂ ϕ˜ (a)(zj − aj )(zk − ak ) + o(|z − a|2 ) ∂zj ∂ z¯k = 2Re(pa (z)) + n j,k=1 ∂ ϕ˜ (a)(zj − aj )(zk − ak ) + o(|z − a|2 )✳ ∂zj zk ỏ ữợ t tr Y tỗ t ởt U ❝õ❛ a s❛♦ ❝❤♦ ϕ(z) ˜ ≥ Re(2pa (z)) = log |e2p (z) |, ∀z ∈ U ✳ ❚ø ✤â s✉② r❛ ϕ ❝â t➼♥❤ ❝❤➜t (S) t↕✐ a✳ ❚❤❡♦ ❇ê ✤➸ 2.9 t❛ ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ a ✶✾ ❍➺ q✉↔ ✷✳✶✷✳ ❬✶✶❪✳ ●✐↔ sû X ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❤②♣❡r❜♦❧✐❝ ✤➛②✱ ϕ ❧➔ ♠ët ❤➔♠ ❧✐➯♥ tư❝ ❝â ❣✐→ trà t❤ü❝ tr➯♥ X ✳ ●✐↔ t❤✐➳t r➡♥❣ ợ z0 X tỗ t ởt ❧➙♥ ❝➟♥ U ❝õ❛ z0 s❛♦ ❝❤♦ ϕ(z) = supj≥1 {cj log |fj (z)|}, ∀ z ∈ U, (1) tr♦♥❣ ✤â {fj }j≥1 t❤ä❛ ♠➣♥ log |fj | > tr➯♥ U ✈ỵ✐ ♠å✐ j ❧➔ ♠ët ❞➣② ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ U ✈➔ {cj }j≥1 ❧➔ ♠ët ❞➣② ❝→❝ sè ❞÷ì♥❣ t❤ä❛ ♠➣♥ < a < cj < b < ∞, ∀ j ❑❤✐ ✤â Ωϕ (X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤å♥ ♠ët ❞➣② ❝♦♥ {nj }j≥1 s❛♦ ❝❤♦ ϕ(z0 ) = limj→∞ cnj log |fnj (z0 )| ❱➻ ϕ ❧➔ ❜à ❝❤➦♥ tr➯♥ U s❛✉ ❦❤✐ t❤✉ ♥❤ä U ✱ tø ✭✶✮ t❛ ❝â supj≥1 ||fnj ||U ≤ supU ϕ < ∞ ❱➻ ✈➟②✱ ❜➡♥❣ ❝→❝❤ ❧➜② ❞➣② ❝♦♥ ✈➔ t❤✉ ♥❤ä U ❧➛♥ t❤ù ❤❛✐✱ ❝â t❤➸ t❛ ❣✐↔ t❤✐➳t r➡♥❣ ❞➣② {fn }j1 tử ữỡ tr U tợ ởt ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ f ✳ ❉♦ ✤â cj → c ≥ a > 0✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ j ϕ∗ (z) = c log |f (z)|, ∀z ∈ U ❉➵ ❦✐➸♠ tr❛ ✤÷đ❝ ϕ∗(z0) = ϕ(z0) ✈➔ ϕ∗(z) ≤ ϕ(z), ∀z ∈ U ✳ ❱➻ ✈➟② ϕ ❝â t➼♥❤ ❝❤➜t ✭❙✮ t↕✐ ✤✐➸♠ z0 ✳ ❚❤❡♦ ❇ê ✤➲ 2.9✱ t❛ ✤÷đ❝ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â t❤➸ ❝❤➾ r❛ ❝â ỏ ữợ ổ õ t t (S) t↕✐ ♠ët ✤✐➸♠✳ ✷✵ ◆❤➟♥ ①➨t ✳ ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝✱ ϕ ❧➔ ❤➔♠ ✤❛ ỏ ữợ tr X sỷ r ❝â t➼♥❤ ❝❤➜t ✭❙✮ t↕✐ ✤✐➸♠ z0 ∈ X ✱✈➔ ϕ(z0 ) = −∞✱ t❛ s➩ ❝❤ù♥❣ tä r➡♥❣ ♠✐➲♥ X ∗ = (z, w) ∈ X × C : Re w + (z) < tọ tỗ t ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ z0✱ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ ψ tr➯♥ U ♠➔ X ∗ ∩ {(z, ψ(z))|z ∈ U } = ∅✱ tr♦♥❣ ✤â ψ(z0) = −ϕ(z0)✳ ❚❤➟t ✈➟②✱ ❧➜② ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ z0 tr♦♥❣ X ✤õ ❜➨ s❛♦ ❝❤♦ U ❧➔ ❝♦ rót ✤÷đ❝ ❳✳ ❱➻ ϕ(z0) = −∞ ♥➯♥ t❛ ❝â fz (z0) = s r tỗ t ởt tr➯♥ U t❤ä❛ ♠➣♥✿ |fz0 (z)eψ(z) | = 1|, ψ(z0 ) = −ϕ(z0 ), ∀z ∈ U ❚❛ ❝❤ó þ r➡♥❣ ϕ(z) ≥ log |fz0 (z)| = log |e−ψ(z) | = −Re ψ(z), ∀ z ∈ U, ✈➔ ψ(z0) = −ϕ(z0)✳ ❱➻ ✈➟② t❛ ✤÷đ❝ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❚❤❡♦ ✈➼ ❞ö ❝õ❛ ❑♦♥♥ ❛♥❞ ◆✐r❡♥ ❜❡r❣ ✭❬✶✷❪✮ ✈➔ tứ t tr t s r tỗ t ởt ỏ ữợ tr tỹ tr C2 s❛♦ ❝❤♦ ϕ ❦❤æ♥❣ ❝â t➼♥❤ ❝❤➜t (S) t↕✐ s ự tọ r tỗ t ởt ❤➔♠ ϕ s❛♦ ❝❤♦ Ωϕ(X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛② ♥❤÷♥❣ ϕ ❦❤æ♥❣ ❝â t➼♥❤ ❝❤➜t (S) t↕✐ ♠ët ✤✐➸♠✳ ▼➺♥❤ ✤➲ ✷✳✶✸✳ ❬✶✶❪✳ ●✐↔ sû X ❧➔ ♠ët ♠✐➲♥ ❤②♣❡r❜♦❧✐❝ ✤➛② tr♦♥❣ C ❝❤ù❛ 0, ❧➜② t > s❛♦ ❝❤♦ cos(5π/7) < −1/t < −5/9✳ ❳→❝ ✤à♥❤ ❤➔♠ ϕ ❜ð✐ ϕ(z) = |z|6 + t|z|2 Re(z ) ✷✶ ❑❤✐ ✤â Ωϕ (X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛② ♥❤÷♥❣ ϕ ❦❤ỉ♥❣ ❝â t➼♥❤ ❝❤➜t ✭❙✮ t↕✐ 0✳ ▼➺♥❤ ✤➲ ♥➔② ✤÷đ❝ s✉② trü❝ t✐➳♣ tø ❤❛✐ ❇ê ✤➲ s❛✉ ✿ ❇ê ✤➲ ✷✳✶✹✳ ❬✶✶❪✳ ●✐↔ sû X ❧➔ ♠ët ♠✐➲♥ ❤②♣❡r❜♦❧✐❝ ✤➛② tr C, ởt ỏ ữợ tr X ✳ ●✐↔ t❤✐➳t r➡♥❣ t↕✐ ♠å✐ ✤✐➸♠ z0 ∈ X tỗ t ởt số tỹ n 4✱ ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ z0 s❛♦ ❝❤♦ ϕ ❧➔ ❤➔♠ t❤✉ë❝ ❧ỵ♣ C n tr➯♥ U ✱ ✈➔ ỡ ỳ tỗ t i, l s i ≤ l − ≤ n − ✈➔ ∂l ϕ (z0 ) = ∂z l−i ∂ z¯ (2) ❑❤✐ ✤â t❛ ❝â Ωϕ (X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ❈❤ù♥❣ ♠✐♥❤ ▲➜② (z0, w0) ∈ ∂Ωϕ(X) s❛♦ ❝❤♦ z0 ∈ X ✈➔ |w0eϕ(z )| = ❱ỵ✐ r > ✤õ ❜➨✱ t❛ ①→❝ ✤à♥❤ ✭✸✮ Wz0 , r = {(z, w) ∈ X × C : |z − z0 | < r, |w| < e−ϕ(z) } ⊂ Ωϕ (X) ❳➨t ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ❝õ❛ ϕ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ✤õ ❜➨ ❝õ❛ z0 ϕ(z) = nj=0 Pj (z − z0 ) + o(|z − z0 |n ), tr♦♥❣ ✤â Pj ❧➔ ✤❛ t❤ù❝ t❤✉➛♥ ♥❤➜t ❝õ❛ z, z¯ ✈➔ ❝â ❜➟❝ j ✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ Pl ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛✳ ❚❛ ✈✐➳t l ❚❛ ❝â Pl (z) = ∆Pl (z) = l m=0 ∂ϕ ∂z l−m ∂ z¯ m (z0 )z l−m z¯ m ∂lϕ − m) l−m m (z0 )z l−m−1 z¯ m−1 ∂z ∂ z¯ ∆Pl ✈➔ ✤✐➲✉ ❦✐➺♥ ✭✷✮ s✉② r❛ Pl ❦❤æ♥❣ l−1 m=1 m(l ❚ø ❜✐➸✉ t❤ù❝ ♥➔② ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✳ ❱➻ ✈➟②✱ t❛ õ t t ữợ s (z) = ψ(z) + Pk (z − z0 ) + o(|z − z0 |k ), t❤➸ ❧➔ (4) tr♦♥❣ ✤â ψ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ✈➔ k ∈ [1, n] ❧➔ sè ♥❣✉②➯♥ ❜➨ ♥❤➜t s❛♦ ❝❤♦ Pk ❦❤æ♥❣ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛✳ ✷✷ ▲➜② ψ˜ ❧➔ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ z0 t❤ä❛ ♠➣♥✿ ˜ Re(ψ(z)) = ψ(z) ❚❛ ✈✐➳t ˜ |w2 e2ϕ(z) | = |weψ(z) |2 e2(ϕ(z)−ψ(z)) = |weψ(z) |2 e2(ϕ(z)−ψ(z)) ❉ò♥❣ ♣❤➨♣ ✤è✐ tå❛ ✤ë z = z − z0 ˜ ˜ w = weψ(z) − w0 eψ(z0 ) ❑❤✐ ✤â Wz ,r s➩ ✤÷đ❝ ❜✐➳♥ ✤è✐ t❤➔♥❤ ˜ Wr = {(z , w ) : |z | < r, |w + w0 eψ(z0 ) |2 |e2(ϕ(z +z0 )−ψ(z +z0 )) | < 1} ˜ k = {(z , w ) : |z | < r, |w + w0 eψ(z0 ) |2 e2 Re Pk (z )+o(|z | ) < 1} = (z , w ) : |z | < r, ˜ (1 + |w |2 + Re w w¯0 e−ψ(z0 ) + Re Pk (z ) + o(|z |k ) < ˜ 0) −ψ(z = (z , w ) : |z | < r, + |w |2 + Re w w¯0 +2 Re Pk (z ) + |w |2 ˜ +4 Re Pk (z )Re w w¯0 e−ψ(z0 ) +o(|z |k ) < ˜ = (z , w ) : |z | < r, Re Pk (z )+2 Re w w¯0 e−ψ(z0 ) +o(|w |+|z |k ) < Ð ✤➙②✱ ❞á♥❣ t❤ù ❤❛✐ ❝â tø ✭✹✮ ✈➔ ❞á♥❣ t❤ù ❜❛ ❝â tø ✭✸✮ ✈➔ tø ❦❤❛✐ tr✐➸♥ ex = + x + o(x)✳ ❱➻ ✈➟② ✈ỵ✐ r > 0, ε > ✤õ ❜➨✱ Wr ❜à ❝❤ù❛ tr♦♥❣ ♠✐➲♥ Ωε = (z , w ) : |z | < r, Re w + Re Pk (z ) < |ε|(|w | + |z |k ) ỵ ỡ ❝õ❛ ❇❡❞✲❢♦r❞ ✈➔ ❚❛②❧♦r tr♦♥❣ ([4], p559) ❝❤ó♥❣ t❛ ❝â ♠ët ❤➔♠ f ∈ Hol(Ωε) ∩ C(Ω¯ε) s❛♦ ❝❤♦ f (0, 0) = > |f (z , w )|, (z , w ) ứ ỵ ✷✳✼ t❛ ❝â Ωϕ(X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ❱➟② ❜ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✸ ❇ê ✤➲ ✷✳✶✺✳ ❬✶✶❪✳ ❈❤♦ X ❧➔ ♠ët ♠✐➲♥ ❤②♣❡r❜♦❧✐❝ ✤➛② tr♦♥❣ C ❝❤ù❛ 0✱ ❧➜② t > s❛♦ ❝❤♦ cos 5π < − t < − ✳ ❱ỵ✐ ♠å✐ sè m tỗ t (0, 2) s❛♦ ❝❤♦ + t cos(4θ0 ) < 0, cos(mθ0 ) > ❈❤ù♥❣ ♠✐♥❤✿ ❱➻ cos(5π/7) < −1/t < 5/9 tỗ t (/2, 5/7) s tọ ♠➣♥✿ ❝❤♦ cos α = −1/t✳ ❚❛ s➩ ❝❤➾ r❛ tỗ t m4 , m(2) cos( ) > 0✳ ❚❤➟t ✈➟②✱ t❛ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣✿ ♥➳✉ m ≥ 7, t❛ ❝â t❤➸ ❝❤å♥ 6(2π−α) 3π ✤÷đ❝ θ ✈➻ m(2π−α) − mα − = 3(π−α) > ✈➔ 4 > π ✱ ♥➳✉ m = t❤➻ ✈➻ m(2π−α) 3π 6(2π−α) < 5π t❤ä❛ ♠➣♥ cos(θ ) > 0✳ ♥➯♥ t❛ ❝â t❤➸ ❝❤å♥ θ ∈ , ❇➡♥❣ ❝→❝❤ ❝❤å♥ θ0 = θ /m t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❈❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ✷✳✶✸✳ ❱➻ −1/t < −5/9 ♥➯♥ t❛ ❝â ∆ϕ(z) = 9|z|4 + 5t(z + z¯ )/2 ≥ 0, ∀z ∈ C ❱➻ ✈➟② t tỹ ỏ ữợ tr C ú ỵ r t z0 C, ✤✐➲✉ ❦✐➺♥ ✭✷✮ ❧➔ ✤÷đ❝ t❤ä❛ ♠➣♥ ✭♥â✐ ❝→❝❤ ❦❤→❝✱ t➜t ❝↔ ✤↕♦ ❤➔♠ r➯♥❣ ❝õ❛ ∆ϕ s➩ tr✐➺t t✐➯✉ t↕✐ z0✱ ✈➔ ❤➔♠ ❣✐↔✐ t➼❝❤ t❤ü❝ ∆ϕ ♣❤↔✐ ❧➔ ỗ t tr ởt z0 ♥➔② rã r➔♥❣ ❧➔ ❦❤æ♥❣ t❤➸✮✳ ❱➻ ✈➟② t➜t ❝↔ ❝→❝ ❣✐↔ t❤✐➳t tr♦♥❣ ❇ê ✤➲ ✷✳✶✹ ✤÷đ❝ t❤ä❛ ♠➣♥✱ ✈➔ ❞♦ ✈➟② Ωϕ (X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ❚❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ϕ ❦❤æ♥❣ ❝â t➼♥❤ ❝❤➜t ✭❙✮ t sỷ ữủ õ tỗ t ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ f tr➯♥ ✤➽❛ U ❝❤ù❛ s❛♦ ❝❤♦ ϕ(z) ≥ log |f (z)|, ∀z ∈ U, = log |f (0)| (5) ❘ã r➔♥❣ t❛ ❝â t❤➸ ✈✐➳t f = eg tr➯♥ U ✱ ✈➻ ✈➟② ✭✺✮ ✤÷đ❝ ✈✐➳t ❧↕✐ ❧➔ ϕ(z) ≥ Re g(z) , ∀z ∈ U, Re g(0) = ✷✹ ●✐↔ sû ϕ < tr➯♥ ♥û❛ ✤÷í♥❣ t❤➥♥❣ l = {reiπ/4 : r > 0}✱ g ❝â t❤➸ ❦❤æ♥❣ ❧➔ ❤➡♥❣ sè✳ ❇➡♥❣ ❝→❝❤ ❦❤❛✐ tr✐➸♥ g ✈➔♦ ❝❤✉é✐ ❚❛②❧♦r ❝❤ó♥❣ t❛ ✤↕t ✤÷đ❝ ϕ(z) ≥ Re az m + o(|z|m ) , a = 0, ∀z ∈ U (6) ◗✉❛ ♠ët ♣❤➨♣ q✉❛②✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ a = 1✳ ❱➻ ✈➳ tr→✐ ❝õ❛ ✭✻✮ ❝â ❜➟❝ ❜➡♥❣ ✻✱ ♥➯♥ tø ✭✻✮ t❛ ❝â t❤➸ ❦➳t ❧✉➟♥ r➡♥❣ m ≥ 6✳ ▲➜② θ0 ❧➔ ♠ët sè t❤ä❛ ♠➣♥ ❇ê ✤➲ ✷✳✶✹✳ ❑❤✐ ✤â tø ❝→❝❤ ❝❤å♥ ❝õ❛ θ0✱ t❛ ❝â ϕ(z) < 0, ∀z ∈ l = {reiθ0 : r > 0} ▼➦t ❦❤→❝✱ ✈➻ Re(zm) = rm cos(mθ0), ∀z ∈ l ✱ ❝❤ó♥❣ t❛ ❦➳t ❧✉➟♥ r➡♥❣ ữỡ ợ z l ổ ỵ ✈➟② ϕ ❦❤æ♥❣ t➼♥❤ ❝❤➜t (S) t↕✐ 0✳ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✺ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ t➼♥❤ ❤②♣❡r✲ ❜♦❧✐❝ ✤➛② ❝õ❛ ♠✐➲♥ ❍❛rt♦❣s Ωϕ(X)✱ ❝ö t❤➸ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉✿ +) ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝✱ ϕ : X → [−∞, ∞) ❧➔ ❤➔♠ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ ①→❝ ✤à♥❤ tr➯♥ X ✳ ◆➳✉ Ωϕ(X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✱ t❤➻ X ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛② ✈➔ ϕ ❧➔ ♠ët ❤➔♠ ❣✐→ trà tỹ ỏ ữợ tử tr X +) ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ ❤②♣❡r❜♦❧✐❝✱ ởt ỏ ữợ tử ❣✐→ trà t❤ü❝ tr➯♥ X t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t s❛✉✿ ❱ỵ✐ ♠é✐ ✤✐➸♠ ❜✐➯♥ (z0, w0) ❝õ❛ Ωϕ(X) ✈ỵ✐ z0 X tỗ t ởt V z0 tr♦♥❣ X ✈➔ ♠ët →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ f tø Ωϕ (X) ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ ❤②♣❡r❜♦❧✐❝ ✤➛② Y s❛♦ ❝❤♦ ❞➣② f (zn, wn) ❦❤ỉ♥❣ ❧➔ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ tr♦♥❣ Y ✈ỵ✐ ❜➜t ❦ý ❞➣② {(zn, wn)} ❤ë✐ tư tỵ✐ (z, w)✳ ❑❤✐ ✤â Ωϕ(X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ +) ●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ●✐↔ tt r ởt ỏ ữợ ♥❣➦t tr➯♥ X ❑❤✐ ✤â Ωϕ(X) ❧➔ ❤②♣❡r❜♦❧✐❝ ✤➛②✳ ✷✻ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪✳ P❤↕♠ ❱✐➺t ✣ù❝✱ ỵ tt ổ ự r ◆❳❇ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✱ ✷✵✵✺✳ ❬✷❪✳ P❤↕♠ ❱✐➺t ✣ù❝✱ ❚➼♥❤ ❤②♣❡r❜♦❧✐❝ ✤➛② ✈➔ t➼♥❤ ♥❤ó♥❣ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤ù❝✱ ❧✉➟♥ →♥ t✐➳♥ s➽ t♦→♥ ❤å❝✱ ✷✵✵✵✳ II ❚✐➳♥❣ ❆♥❤✳ ❬✸❪✳ ❚✳❏✳ ❇❛rt❤✱ ❚❤❡ ❑♦❜❛②❛s❤✐ ❞✐st❛♥❝❡ ✐♥❞✉❝❡s t❤❡ st❛♥❞❛r❞ t♦♣♦❧♦❣②✱ ♣r♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✱ ✸✺✱ ✹✸✾✲✹✹✶✱ ✶✾✼✷✳ ❬✹❪✳ ❊✳ ❇❡❣❢♦r❞ ❛♥❞ ❏✳ ❋♦r♥❛❡ss✱ ❆ ❝♦♥str✉❝t✐♦♥ ♦❢ ♣❡❛❦ ❢✉♥❝t✐♦♥ ♦♥ ✇❡❛❦❧② ♣s❡✉❞♦❝♦♥✈❡① ❞♦♠❛✐♥✱ ❆♥♥✳ ▼❛t❤✱ ✶✵✼✱ ✺✺✺✲✺✻✽✱ ✶✾✼✽✳ ❬✺❪✳ ❏✳ ❋♦r♥❛❡ss ❛♥❞ ❘✳ ◆❛r❛s✐♠❤❛♥✱ ❚❤❡ ▲❡r✐ ♣r♦❜❧❡♠ ♦♥ ❝♦♠♣❧❡① s♣❛❝❡s ✇✐t❤ s✐♥❣✉❧❛r✐t✐❡s✱ ❆♥♥✳ ▼❛t❤✱ ✷✹✽✱ ✹✼✲✼✷✱ ✶✾✽✵✳ ❬✻❪✳ P✳❑✐❡r♥❛♥✱ ❖♥ t❤❡ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❛✉t✱ t✐❣❤t ❛♥❞ ❤②♣❡r❜♦❧✐❝ ♠❛♥✐❢♦❧❞s✱ ❇✉❧❧✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✱ ✼✻✱ ✹✾✲✺✶✱ ✶✾✼✵✳ ❬✼❪✳ ❙✳ ❑♦❜❛②❛s❤✐✱ ❍②♣❡r❜♦❧✐❝ ❝♦♠♣❧❡① s♣❛❝❡s✱ ❱ ✸✶✽✳ ●r✉♥❞❧❡❤r❡♥ ❞❡r ♠❛t❤❡♠❛t✐s ❝❤❡♥ ❲✐ss❡♥s❝❤❛❢t❡♥✱ ✶✾✾✽✳ ❬✽❪✳ ❙✳ ▲❛♥❣✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❝♦♠♣❧❡① ❍②♣❡r❜♦❧✐❝ s♣❛❝❡s✱ ❙♣r✐♥❣❡r ✲ ❱❡r❧❛❣✱ ◆❨✳ ❬✾❪✳ ❚❤✳ P❡t❡r♥❡❧❧✱ Ps❡✉❞♦❝♦♥✈❡①✐t②✱ t❤❡ ▲❡r✐ ♣r♦❜❧❡♠ ❛♥❞ ✈❛♥✲ ✐s❤✐♥❣ t❤❡♦r❡♠s✱ ❊♥❝②❝❧♦♣❛❡❞✐❛ ♦❢ ▼❛t❤✳ ❙❝✐❡♥❝❡s✱ ✼✹✱ ✸✺✼✲✸✼✷✱ ✶✾✾✹✳ ❬✶✵❪✳ ❉✳❉✳ ❚❤❛✐ ❛♥❞ P✳❱✳ ❉✉❝✱ ❖♥ t❤❡ ❝♦♠♣❧❡① ❤②♣❡r❜♦❧✐❝✐t② ❛♥❞ t❤❡ t❛✉t♥❡s ♦❢ t❤❡ ❍❛rt♦❣s ❞♦♠❛✐♥s✱ ■♥t❡r✳ ❏♦✉r✳ ▼❛t❤✱ ✶✶✱ ✶✵✸ ✲ ✶✶✶✱ ✷✵✵✵✳ I ✷✼ ❬✶✶❪✳ ❉✳ ❉✳ ❚❤❛✐ ❛♥❤ ◆✳ ◗✳ ❉✐❡✉✱ ❈♦♠♣❧❡t❡ ❤②♣❡r❜♦❧✐❝❝✐t② ♦❢ ❍❛r✲ t♦❣s ❞♦♠❛✐♥s✱ ▼❛♥✉s❝r✐♣t❛ ♠❛t❤✱ ✶✶✷✱ ✶✼✶ ✲ ✶✽✶✱ ✷✵✵✸✳ ❬✶✷❪✳ ◆✳ ❙✐❜♦♥②✱ ❙♦♠❡ ❛s♣❡❝ts ♦❢ ✇❡❛❦❧② ♣s❡✉❞♦❝♦♥✈❡① ❞♦♠❛✐♥s✳ Pr♦✲ ❝❡❡❞✐♥❣s ♦❢ ❙②♠♣♦s✐❛ ✐♥ P✉r❡ ▼❛t❤✱ ✺✷✱ ✶✾✾ ✲ ✷✸✶✱ ✶✾✾✶✳ ✷✽